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A in y with <1>(0) E u. But since , p) E C in statements (1) - (4) of Proposition 3.4 and does not cluster to p can be replaced by (, p) does not belong to C. Then we have: THEOREM 3.4 Let C be a cluster class on a set X and for each A eX let CI(A) be the set of all x E X with ( ).. (y) for some A E A. To see that 4> is a uniform homeomorphism, we show that ~h~(x/
To show CI(CI(A» = CI(A) suppose a E CI(CI(A» which implies the existence of a transfinite sequence \jI:Y -t CI(A) with (\jI, a) E C. For each e E Y, \jI(e) E CI(A) which implies there is a transfinite sequence \jIe:Ye ---7 A such that (\jIe,\jI(e» E C. But then the sum L of the transfinite sequences {\jIe} and the point a form a pair (L, a) E C by statement (4) of Proposition 3.4. Therefore a E CI(A) since LeA. Consequently CI(CI(A» c CI(A) and hence CI(CI(A» = CI(A). To show CI(AuB) = CI(A)uCI(B) first note that a E CI(A) implies there is a pair (\jI,a) E C with \jI c A and hence \jI c AuB so a E CI(AuB). Similarly a E CI(B) implies a E CI(AuB) so CI(A)uCI(B) c CI(AuB). Finally, p E CI(AuB) implies the existence of a pair (\jI, p) E C with \jI c AuB. Let M = YIl\jl-l(A) and N = YIl\jl-l(B) where \jI:Y-t X. Then Y= MuN and by statement (2) of Proposition 3.4, either (\jIM,P) or (\jIN,P) E C. Since'VM c A and \jIN c B either P E Cl(A) or P E CI(B). Hence P E CI(A)uCI(B). We conclude that CI(AuB) = CI(A)uCI(B).
3.3 Transfinite Sequences and Topologies
81
It remains to show that ('I', p) E C if and only if 'I' clusters to a relative to the topology associated with Cl which will now be referred to as 'to First suppose ('I', a) E C but 'I' does not cluster to a relative to 'to Then there is an open neighborhood U of a such that 'I' is not frequently in U which implies a residual R c y with 'I'(R) c X - U where 'I':y --t X. By statement (2) of Proposition 3.4, ('I'R, a) E C which implies a E CI(X - U). But this is a contradiction since U is open. Therefore 'I' clusters to a relative to 'to
Conversely, suppose 'I':y --t X clusters to a. Then a E CI(Me) for each e E y where Me = {'I'(~) Ie :<::; ~). Therefore there is a transfinite sequence 'l'e:')'e ---7 Me with ('I'e, a) E C for each e E y. By statement (3) of Proposition 3.4 we must then have ('I', a) E c.Proposition 3.4 and Theorem 3.5 set up a one-to-one correspondence between the various topologies a set can have and the cluster classes on the set. It is clear from the definition of clustering that if C J and C 2 are two cluster classes and 'tJ and't2 are the associated topologies, that C J C C 2 if and only if 't2 C 'tJ. Moreover, it is interesting to note that if (C J nC 2) denotes the smallest cluster class that is larger than each of C J and C 2 then (C J nC 2) is the cluster class associated with 'tJ n'!2' We conclude this chapter with a characterization of continuous functions in terms of transfinite sequences. Various other interesting mappings (e.g., open mappings, closed mappings, quotient mappings, etc.) can be characterized in a similar manner but this will be left for the exercises. PROPOSITION 3.5 Let f:X --t Y be a function from a space X into a space Y. Thenfis continuous if and only iffor each transfinite sequence {xa) in X that clusters to some p E X. {f(x a ) I clusters to f(p). Proof: Assume f is continuous and suppose {x a) is a transfinite sequence in X that clusters to some point p E X. Let U be a neighborhood of f(P) in Y and pick a neighborhood V of p such that fey) c U. Then {x a) frequently in V implies {f(x a )) is frequently inf(V) c U, so {f(x a )) clusters tof(P).
Conversely, assume that for each transfinite sequence {x a) in X that clusters to some point p in X, {f(xa)) clusters to f(P). Suppose f is not continuous. Then there is apE X and a neighborhood U of f(P) such thatf(V) is not contained in U for each neighborhood V of p. By Theorem 3.4 there exists a well ordered neighborhood base {Val a < y) of p such that < is compatible with the partial ordering of set inclusion. For each a E A pick x a E Va such that f(x a) is not contained in U. Then {x a) clusters to p but {f(x a)) does not cluster to f(P) which is a contradiction. We conclude that f must be continuous. -
82
3. Transfinite Sequences
EXERCISES
1. Show that X is T I if and only if for each pair of distinct points in X there are two transfinite sequences clustering to the two points respectively but neither clustering to the other point. 2. A function is said to be open if the image of each open set is again an open set. Show that an onto function f is open if and only if for each transfinite sequence {y ~} in Y that clusters to some p E Y and for each q E rl (P) there is a transfinite sequence {xa} in u{f-l(Yj3)} that clusters to q. 3. A function is said to be closed if the image of each closed set is a closed set. Show that a function f is closed if and only if whenever a transfinite sequence {y ~} clusters to some p E Y there is a transfinite sequence Ix a} that is a subset ofrl({y~}) that clusters tor1(p). 4. A function is said to be a quotient if whenever the inverse image of a set is open, then the set itself must be open. Show that a function f is a quotient if and only if for each transfinite sequence {y~} clustering to some p E Y there is a transfinite sequence {x a} contained in the inverse image of {y /3} under f that is frequently in each open inverse image of an open set in Y that contains r 1(P). 5. Show that Corollary 3.2 still holds when the assumption of complete regularity is removed from the hypothesis; i.e., show that any space is compact if and only if each transfinite sequence clusters.
Chapter 4 COMPLETENESS, COFINAL COMPLETENESS AND UNIFORM PARACOMPACTNESS
4.1 Introduction In 1915, A paper by E. H. Moore appeared in the Proceedings of the National Academy of Science U.S.A. titled Definition of limit in general integral analysis. This study of unordered summability of sequences led to a theory of convergence by Moore and H. L. Smith titled A general theory of limits which appeared in the American Journal of Mathematics in 1922. In 1937, G. Birkhoff applied the Moore-Smith theory to general topology in an article titled Moore-Smith convergence in general topology, which appeared in the Annals of Mathematics, No. 38, pp. 39-56. In 1940, J. W. Tukey made extensive use of the theory in his monograph titled Convergence and uniformity in topology published in the Annals of Mathematics Studies series. Tukey worked with objects that were generalizations of sequences that he referred to as phalanxes. They were a special case of the objects that are usually called nets today. An equivalent theory of convergence using objects called filters emerged in the thirties from the Bourbaki group in France. The theory of filters is the convergence theory of choice for many topologers. There are many things to recommend it but it is also very awkward to use in certain situations. For instance, in the treatment of hyperspaces (Chapter 5), the objects of the hyperspace are subsets of the original space. Filters, which are themselves collections of subsets of a space, become awkward to construct in the hyperspace because they are collections of collections of subsets. For some proofs, filters carry along too much baggage. In proofs about the completions of uniform spaces it is sometimes desirable to pick a convergence object from the original space that is arbitrarily close to a convergence object in the completion. Then conclusions are made about the convergence object in the original space and these conclusions are then shown to hold for the convergence object in the completion due to its proximity to the object in the original space. A natural way to attempt this with filters is to restrict each subset in the completion filter to the original space, make conclusions about the restricted filter, then take the closures of the members of the restricted filter to get back to the completion. The problem here is that taking the closure picks up too many
84
4. Completeness, Cofinal Completeness and Uniform Compactness
points. Since the original filter in the completion is not always obtained, it may not be possible to draw conclusions about the original filter in the completion. Surely with time, one should be able to find filter type proofs for the theorems we will be presenting in the sequel, but this will not be our viewpoint. Instead, in some of the areas where the filter type proofs are especially nice, the recasting of the results into filter terminology will be suggested as exercises.
4.2 Nets A non-void set D is said to be directed by the binary relation the following conditions hold:
~
provided that
(1) if m,n and d E D with m ~ nand n ~ d then m ~ d, (2) d ~ d for each d in D and (3) if m,n E D then there exists a d in D with m ~ d and n ~ d.
Clearly a directed set is a particular type of partially ordered set. Condition (3) in the definition of a directed set is what sets the directed set apart from an ordinary partially ordered set. Also, it should be noticed that well ordered sets are necessarily directed sets. Some useful examples of directed sets that are not necessarily well ordered are given below. Let X be a topological space and let B be a local neighborhood base for a point p E X. Define the relation ~ on B as follows: U ~ V if U,V E B and V c U. Clearly (B, ~) satisfies conditions (1) and (2) above so (B, ~) is a partially ordered set. To show (B,~) also satisfies (3) assume U,V E B. Then W = UnV belongs to Band W c U and W c V. Thus U ~ Wand V ~ W, so (B,~) satisfies (3).
Another directed set that we will employ frequently is the uniformity itself. Let (X. f.!) be a uniform space. The relations < (refinement) and <* (star refinement) have already been defined for f.! (see Section 2.1). Clearly (f.!, <) and (11, <*) satisfy conditions (1) and (2) defined above so that both are partially ordered sets. That < also satisfies condition (3) above follows from condition (1) of Section 2.1. At this point we are careful to remind the reader that we consider these orderings as "proceeding toward the left" in the sense that in proving condition (3) for <* we will show that if U,V E 11 then there exists aWE 11 such that W* < U and W* < V. By (1) of Section 2.1 there exists a Z E 11 such that Z < U and Z < V. By (3) there is a Win 11 with W <* Z. Therefore W <* U and W <* V. Consequently both (11, <) and (11, <*) are directed sets. A subset R 'F- 0 of a directed set (D, ~) is said to be residual if whenever m,n E D with mER and m ~ n then n E R. IfC c D such that whenever m E D there is an n E C with m ~ n we say C is cofinal in D. A net is a function \jI:D
4.2 Nets
85
-t X from a directed set D into a space X. In the event D is well ordered then '" is simply a transfinite sequence with which we are already familiar. For each ex E D let x a = ",(ex). We will often identify a net", with its range ",(D) = {x a I ex E D). If the set D is understood we simply write {xa)' We say a net {xa) is
frequently in U c X if there is a cofinal C cD with {x~ I ~ E C) cU. We say {x a) is eventually in U if there is a residual ReD with (x y I 'Y E R) cU. {x a) is said to converge to a point p in X if it is eventually in each neighborhood of p and to cluster to p if it is frequently in each neighborhood of p. PROPOSITION 4.1 A subset U of a space X is open net in X - U converges to a point of U.
if and only if no
Proof: Assume U is open in X and let {x a) be a net that converges to some p E U. From the definition of convergence {x a) must eventually be in U. But then {x a) cannot lie entirely in X - U. Conversely, assume no net in X - U converges to a point of U and suppose U is not open. Then there is apE U every neighborhood of which meets X - U. Let D be a local basis for p and let :s; be the partial ordering of set inclusion on D. We have already seen that D is directed with respect to this ordering. For each V E D pick Xv E Vn(X - U). Then {xv) is a net in X - U. It is easily shown that {xv) converges to p. Indeed, let WED and put R(W) = {V E D IV c W). Then R(W) is residual in D and Xv E W for each V E R(W). But this is a contradiction since {xv} lies in X - U. We conclude that U must be open. PROPOSITION 4.2 A point p of a space X belongs to the closure of E c X
if and only if there is a net in E that converges to p.
Proof: Assume p E CI(E) and let D be a local basis for p. Then D is directed by the partial ordering of set inclusion. For each V E D pick Xv E V nE. Then {vv) is a net in E. An argument similar to the one in Proposition 4.1 shows that {xv} converges to p. Conversely, assume (x a) is a net in E converging to a point p E X. Let U be an open set containing p. Then {x a ) is eventually in U which implies U nE *" 0. Thus p E CI(E). PROPOSITION 4.3 A point p of a space X is a limit point of E c X and only if there is a net in E - {p) that converges to p.
if
Proof: Let p be a limit point of E c X and let D be a local basis for p. For each V E D pick Vv E VneE - {p). Then {xv) is a net in E - {p) converging to p. Conversely, if {x a} is a net in E - {p) converging to p E X then for each open set U containing p, {x a) is eventually in U. But then Un(E - {p) *" 0 so p is a limit point of E. -
86
4. Completeness, Cofinal Completeness and Uniform Compactness
PROPOSITION 4.4 A space X is Hausdorff if and only converges to at most one point.
if each net in X
Proof: Let X be a Hausdorff space and assume p and q are distinct points of X. Since X is Hausdorff, there are disjoint open sets U and V such that p E U and q E Y. If {x a} is a net in X that converges to p then {x a} is eventually in U so it cannot eventually be in Y. Thus {x a } cannot converge to q. We conclude that a net can converge to at most one point.
Conversely, assume each net in X can converge to at most one point and suppose X is not Hausdorff. Then there exists two distinct points p and q such that for each pair of neighborhoods U of p and V of q, Un V oF 0. Let B(P) and B(q) be local bases for p and q respectively and put D = B(P) x B(q). Define $ on D as follows: for each a,c E D where a = (A, B) and c = (U,V), put a $ c if U c A and V c B. Then (D, $) is a directed set. For each c = (U,V) E D pick Xc E Un Y. Then {xc} is a net in X. To show that {xc} converges to both p and q let U E B(P) and V E B(q). Put d = (U,V) and let R(d) = {a E Did $ a}. Then R(d) is residual in D. If a E R(d) then a = (A, B) for some A E B(P) and B E B(q) such that A c U and BeY. Moreover, Xa E AnB cUnY. Therefore, for each a E R(d), Xa E U and Xa E V, so {xa} is eventually in both U and V. We conclude {xa} converges to both p and q which is a contradiction. Hence X must be Hausdorff. A net <jl:E ~ X is called a subnet of the net 'II:D function A:E ~ D such that:
~
X if there exists a
(l) <jl = 'V © A and (2) for each d E D there is an e E E such that if e $ c then d $ A(e).
Such a function A is called a cofinal function. It is easily shown that if C is a cofinal subset of D that the identity function i:C ~ D is a cofinal function and hence a net 'II:D ~ X restricted to a cofinal subset C is a subnet of'll. However, in the theory of convergence of nets, we will see that such subnets are usually not very useful. Normally we will be interested in subnets whose domains are not subdomains of the domain of the original net. It should be noted that a subnet is a more complex object than a subsequence of a transfinite sequence and that a subsequence of a transfinite sequence is a sub net when the transfinite sequence is considered as a net. The principle advantage of nets is that one works with convergence rather than clustering (compare Proposition 4.4 with Proposition 3.3).
PROPOSITION 4.5 If B is a family of subsets of the space X with the property that the intersection of two members of B contains a member of Band if the net {x a) is frequently in each member of B then {x a} has a subnet that is eventually in each member of B.
87
4.2 Nets
Proof: Let {xa I a E A} be the net in the hypothesis above. Since the intersection of any two members of B contains a member of B, B is directed hy set inclusion. Then E = {(a, F)
E
A x B Ix a
E
F}
is directed by the ordering :c; defined as follows: if (aI, F I) and (a2' F 2) E E then (a2, F 2) :c; (aI, F I) if a2 :c; al and FIe F 2. Now define a function A:E ~ A by A(a, F) = a for each element (a, F) of E. Clearly A is cofinal so the net lYe leE E} defined by Ye = xA(e) for each e E E is a subnet of {x a }. Next let H E B. Since {xa} is frequently in H, there exists apE A such that x~ E H. Hence e' = (P, H) E E. Let e = (a, F) be an arbitrary member of E such that e' :c; e. Then P:c; a and F c H so Ye
= XA(a,F) = Xa E
F c H.
Thus {x a} is eventually in H.PROPOSITION 4.6 A net clusters to a point subnet that converges to the point.
if and
only
if it
has a
Proof: Let p be a cluster point of the net {x a Ia E A} and let B be a local basis for p. Then the intersection of any two members of B contains a member of B and {xa} is frequently in each member of B. Hence by Proposition 4.5, there exists a sub net {y~} of {xa} that is eventually in each member of B. Hence {y~} converges to p.
Conversely, assume that the net {x a } has a subnet {y~} that converges to p. Let D be the domain of {y ~} and A the cofinal function from D to A that defines {Y~}. Let U he a neighborhood of p and let () E A. Since A is cofinal there is an element E E 0 such that A(P) ~ () for each P ~ E. Since {y~} converges to p there is an element E' ~ E with YE' E U. Now let ()' = A(E'). Then we have ()' ~ () and Xli' =XA(E') = Yf' E U. Hence {xa} clusters to p.PROPOSITION 4.7 A space is compact convergent subnet.
if and only if each
net has a
Proof: By Proposition 4.6 it suffices to show that a space is compact if and only if each net clusters. For this let x = {x a I a E A} be a net in the compact space X. For each a E A let M a be set set of all x~ such that p ~ a. Since A is directed by ~, {Mal a E Al has the finite intersection property so the family {CI(M a) Ia E A I also has the finite intersection property. Since X is compact, by Proposition 0.35, there is apE X which belongs to CI(M a) for each a E A. We will show that x clusters to p. For this let U be a neighborhood of p and let a E A. Since p E CI(M a), it follows that ManU "# 0. Hence there is a f3 E A
88
4. Completeness, Cofinal Completeness and Uniform Compactness
with ~ ~ a and x~ E U. This implies x is frequently in U. We conclude x clusters to p. Conversely, assume every net in X clusters. Let F be a family of closed sets in X with the finite intersection property. Let G denote the family of finite intersections of members of F. Then G also has the finite intersection property. Since F c G, it suffices to show that the intersection of all members of G is nonempty to show that X is compact. Now G is directed by set inclusion c. For each FE G pick Xp E F. Then the assignment F -7 Xp defines a net x = {xp} in X. By hypothesis, x clusters to some p E X. Let H, KEG such that K c H. Then XK EKe H, so x is eventually in the closed set H. By Propositions 4.l and 4.6, p E H. Hence p E H for each HE G. We conclude that X is compact.
-
PROPOSITION 4.8 A function f:X only j(p).
-7
Y is continuous at p
E
X if and
if for each net {x a} in X converging to p. the net (f(x a)} in Y converges to
Proof' Suppose f:X -7 Y is continuous at p and let V be a neighborhood of f(P) in Y. By definition of continuity at p, there is a neighborhood U of p in X such thatfCU) c V. Since {x a } converges to p, there is a residual R in the domain of {x a } such that x~ E U for each ~ E R. Thus f(x~) E feU) c V for each ~ E R, so (f(x a)} converges to f(P). Conversely, assume f is not continuous at p. Then there is an open neighborhood V of f(P) in Y such that every neighborhood of p meets 1 (Y -V). Let B be a local basis for p. Then B is directed by set inclusion. For each U E B pick Xu in Unj-l (Y - V). The assignment U -7 Xu defines a net {xu} in X that converges to p. Now the composition (f(xu)} is a net in Y - V. Since V is open and f(P) E V, (f(xu)} cannot converge to f(P). -
r
In the last chapter, cluster classes of transfinite sequences were discussed and it was shown that the various topologies a space can have correspond to the various cluster classes of transfinite sequences in the space . A similar situation holds for nets, only now it is possible to use convergence instead of clustering. In order to define the concept of a convergence class, it is first necessary to have a result on iterated limits. Iterated limits are usually first encountered in calculus where the limits are the limits of ordinary sequences. Iterated limits of nets can be defined in an analogous manner as in what follows. If (D,
4.2 Nets
89
e if d a
°
Next, consider the case where D is a directed set and for each E D, E 8 is another directed set. Let ~ = {(o,£) 1o E D and £ E E 8}. Consider a function 'I':~ ~ X where X is a topological space. Then for each EDit is possible that the net '1'8 = {'I'(O,£) 1£ E E 8} converges to some point p 8 E X. We also denote p 8 by lima'l'(O,a) and say that the limit of '1'8 exists and equals p 8. Furthermore, it is possible that the net <1> = {p 81o ED} also converges to some point p E X. In this case we say that p is the iterated limit lim8Iima'l'(0,a) of 'I' with respect to and a. Let P = D x fl{E810 ED}.
°
°
°
THEOREM 4.1 Let D be a directed set and for each E Diet E8 be another directed set. Let ~ and P be defined as above. Then there exists a net A:P ~ ~ such that for any function f:~ ~ X for which the iterated limit lim8Iima!(0.a) exists,f© A converges to the iterated limit. Proof: Define A:P ~ ~ as follows: for each (0, d) E P put A(O, d) = (0, d(O». Suppose lim8lima! (o,a) = q and U is an open neighborhood of q. We must find a member (0, d) of P such that if (0, d) < (~, g) then f © A(~, g) E Pick y E D
u.
such that liml3f(~,v) E U for each ~ following y and then, for each such ~ pick d(~) E EI3 such thatf(~.v) E U for all v following d(~) in EI3' If ~ is a member of D which does not follow y let d(~) be an arbitrary member of E 13' If (~, g) > (y, d), then ~ > y, hence liml3f(~,v) E U, and since g(~) > d(~) we have f © A(~,g) = f(~, g(~» E U. We conclude thatf© A converges to q.Let C be a collection of pairs (<1>, p) of nets <1> in the space X and points p E X. We say C is a convergence class for X if it satisfies the following conditions: (1) if <1> is a constant net such that <1>( a) = p for each a then
(<1>, p) E C, (2) if (<1>, p) E C then so does ('I', p) for each subnet 'I' of <1>, (3) if (<1>. p) does not belong to C then there is a subnet 'I' of <1> such that (~, p) does not belong to C for each subnet ~ of 'I' and (4) let D be a directed set and for each E Diet E 8 be another directed set. Let ~ and P and A be defined as in Theorem 4.1. Then for each functionf:~ ~ X for which lim8lima! (o,a) exists, (j© A,p) E C.
°
THEOREM 4.2 Let C be a convergence class for a set X and for each A c X let CI(A) be the set of all p E X with ('I'. p) E C and 'I' c A. Then "Cl" is a closure operator on X and ('I'. p) E C if and only if 'I' converges to p relative to the topology associated with Ct.
The proof of Theorem 4.2 is left as an exercise (see Exercise 2). A net in a space X is said to be a universal net if for each A c X the net is eventually in A
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4. Completeness, Cofinal Completeness and Uniform Compactness
or eventually in X-A. Universal nets have the property that if they are frequently in a set that they must eventually be in the set. Consequently, universal nets converge to each of their cluster points. PROPOSITION 4.9 Every net has a universal subnet. Proof: Let x = Ix 0.1 a. ED} be a net in X and let
(1) x is frequently in each member of
if each
universal nef
Proof: Let X be compact and let let x be a universal net in X. By Propositions 4.6 and 4.7, x must cluster. But a universal net that clusters must also converge. Hence x converges. Therefore compactness implies that each universal net converges. Conversely, if every universal net in X converges, then by Proposition 4.9, every net has a universal subnet so every net in X has a convergent subnet. Then by Proposition 4.7, X must be compact. -
Let (X, f..l.) be a unifonn space. A net {x a.} in X is said to be Cauchy if it is eventually in some sphere of radius U for each U E f..l.. {x a.} is said to be
91
4.2 Nets
cofinally Cauchy if it is frequently in some sphere of radius U for each U E !l. The proofs of the following two propositions are left as exercises (Exercise 3). PROPOSITION 4.10 A net {xa} in a uniform space (X,!l) is Cauchy if and only iffor each U E !l there is a U E U such that {x a} is eventually in U PROPOSITION 4.11 A net {x a J in (X, !l) is cofinally Cauchy if and only iffor each U E !l there is aU E U such that {xa} isfrequently ill U
EXERCISES
1. Let {xa I a. E OJ be a net in the pseudo-metric space (X, d). Show that {xa J converges to p E X if and only if the net {Ya I a. E 0 J, defined by Ya = d(p, x a) for each a. E 0, converges to zero. 2. Prove Theorem 4.2 (see Theorem 3.4).
3. Prove Propositions 4.10 and 4.11. 4. Let {xa Ia. E O} be a net in R (the reals). If a. < ~ in 0 impliesxa ~ XI3 (xa ~ xl3), then {x a } is said to be monotone increasing (decreasing). Show that a
monotone increasing net that is bounded above, or a monotone decreasing net that is bounded below converges. FILTERS 5. A filter on a set X is a collection F of subsets of X satisfying the following properties: (1) F is closed under finite intersections, (2) the empty set does not belong to F, and (3) every subset of X containing a member of F belongs to F. Show that the following three examples of filters satisfy properties (1) through (3) above. The Neighborhood Filter: Let X be a space and let p
E
X. Let F denote the
neighborhood base at p. The Filter Associated with a Net: Let {xa I a. EO} be a net. Put F E F for each ~ E R for some residual ReO}. The Intersection Filter: Let {F a I a. = nF a.
E
= {F c
X IXI3
A J be a non-empty family of filters on X.
Then put F
6. A filter is said to converge to a point p E X if each neighborhood of p belongs to F. F is said to cluster to p if every neighborhood of p meets every member of F. A filter base is a collection B of subsets of X satisfying
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4. Completeness, Cofinal Completeness and Uniform Compactness
properties (1) and (2) in the definition of a filter above. Clearly, the collection F of all subsets of X containing a member of B is a filter called the filter generated by B. The filter base B is said to converge to p E X if the filter generated by B converges to p and to cluster to p if the filter generated by B clusters to p. Show the following: (a) U c X is open if and only if no filter base in X - U converges to a point of U. (b) p E Cl(E) if and only if there is a filter base in E that converges to p. (c) p is a limit point of E c X if and only if there is a filter base in E - {p} that converges to p. 7. Show that a space is Hausdorff if and only if each filter converges to at most one point. 8. Filters can be compared in the following way: Let F and G be two filters on X. F is said to be finer than G and G is said to be coarser than F if G c F. In addition if F :f:. G then F is said to be strictly finer than G and G is strictly coarser than F. Two filters are said to be comparable if one is finer than the other. Show the following: (a) If a filter F clusters to a point p E X then there is a filter G finer than F that converges to p. (b) A space is compact if and only if each filter clusters. (c) A space is compact if and only if each filter is contained in a convergent filter.
4.3 Completeness, Cofinal Completeness and Uniform Paracompactness We have already encountered the concept of completeness in metric spaces (Chapter 1, Section 6). Just as with metric spaces, completeness plays a fundamental role in the theory of uniform spaces. The concept of completeness in metric spaces generalizes in a natural way to uniform spaces. The next three propositions show that Cauchy nets and cofinally Cauchy nets behave in a manner analogous to Cauchy sequences and cofinally Cauchy sequences in metric spaces. Their proofs are also left as exercises (Exercises 1 and 2). PROPOSITION 4.12 A convergent net is Cauchy. PROPOSITION 4.13 A net that clusters is cofinally Cauchy. PROPOSITION 4.14 If a Cauchy net clusters to P then it also converges to p.
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The following theorem is the net version of Theorems 3.1, 3.2 and 3.3. It appeared in the 1971 paper titled On Completeness (Pacific Journal of Mathematics, Volume 38, Number 2, pp. 431-440). The proof is similar to the proofs of Theorems 3.l, 3.2 and 3.3, so we leave it as an exercise (Exercise 6). THEOREM 4.4 (N. Howes, 1971) A space is paracompact (resp. LindelOf or compact) if and only if each net that is cofinally Cauchy with respect to the u (resp. e or~) uniformity clusters.
A uniform space is said to be complete if each Cauchy net converges. It is said to be cofinally complete if each cofinally Cauchy net clusters. COROLLARY 4.1 A cofinally complete uniform space is complete.
The Lebesgue property for metric spaces also generalizes in a natural way to unifonn spaces. We say (X, 11) has the Lebesgue property if for each open covering V of X, there is a U E 11 such that V can be refined by the covering consisting of spheres of radius U. An equivalent characterization of the Lebesgue property is the following: PROPOS1T10N 4.15 (X, 11) has the Lebesgue property if and only if each open covering of X has a refinement in 11 (i.e., 11 is fine and X is paracompact). Proof: Assume each open covering V of X has a refinement say U in 11. Let W <* U. Then the spheres of radius W also refine V so (X, 11) has the Lebesgue property. Conversely, if X has the Lebesgue property and V is an open covering of X, pick U E 11 such that the spheres of radius U refine V. Clearly then U refines V.· PROPOS1T10N 4.16 A compact Hausdorff uniform space has the Lebesgue property. Proof: Let (X, 11) be a compact Hausdorff unifonn space. Then by Theorem 2.7 and Lemma 3.7 we have 11 = ~ is fine. Since X is paracompact, it has the Lebesgue property by Proposition 4.l5.· PROPOSITION 4.17 A uniform space with the Lebesgue property is cofinally complete. Proof: Let (X, 11) be a unifonn space with the Lebesgue property. Suppose (X,ll) is not cofinally complete. Then there exists a cofinally Cauchy net (x u I that does not cluster. For each p E X let U(P) be an open set containing p such that {xu} is eventually in X-U(P) and put U= {U(P)lpE Xl. Since (X, 11) has the Lebesgue property, by Proposition 4.15, U has a refinement V t:: 11. Since
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{x a } is cofinally complete it is frequently in V for some V E V. But V c U(P) for some P E X so {x a} cannot be eventuall y in X - U (P) which is a contradiction. Thus (X, Il) is cofinally complete. •
A uniform space eX, Il) is said to be precompact or totally bounded if each uniform covering has a finite subcovering. Clearly each precompact metric space is a precompact uniform space. THEOREM 4.5 A uniform space is precompact if and only if each net has a Cauchy subnet. Proof: Assume each net in the uniform space (X, Il) has a Cauchy subnet and suppose X is not precompact. Then there is a U E Il such that {S(x, U) I x EX} has no finite subcovering. Pick PI, P 2 E X such that p 2 does not belong to S(p 1, U). Next assume {p 1 . . . Pn} has been defined such that for each pair i,j with i < j ::::; n we have Pj does not belong to S(Pi, U). Since u{S(P 1, U) ... S(pn, U)} cf. X we can choose Pn+l E X with Pn+l not belonging to S(p" U) for each i = 1 ... n. Consequently it is possible (by induction) to construct a sequence {Pn} c X such that for each pair of positive integers i,j with i , j we have Pj does not belong to S(Pi, U).
Let x = {x a I ex ED} be such a Cauchy subnet of the sequence {Pn} for some cofinal function A:D ~ N such that x a = PAra) for each ex E D. Let VEil with V <* U. Since x is Cauchy there is a residual ReD and a POE X such that PAra) E S(p o,v) for each ex E R. Since A is a cofinal function, A(R) is cofinal in N which implies {Pn} is frequently in S(po,v). Pick positive integers k,m with k < m such thatpkoPm E S(Po,V). Then S(PkoV)US(PO,V) c S(Pko U). But then Pm E S(Pko U) which is a contradiction. We conclude (X, Il) is precompact. Conversely, assume (X, Il) is precompact and let x = {x a}, ex E D, be a net in X. If U E Il, then U has a finite subcovering, say lUI ... UN}' Since Ix a} cannot be eventually in X - Uj for j = 1 ... N, it must be frequently in one of them. Therefore, every net in (X, Il) is cofinally Cauchy. By Proposition 4.9, {x a } has a universal subnet IXA(~)} where A:B ~ D is a cofinal function from a directed set B into D. Let VEil. Since IXA(~)} is cofinally Cauchy, there exists a V E V such that {XA(~)} is frequently in V. But then {XA(~)} must eventually be in V, so IXA(~)} is Cauchy.· PROPOSITION 4.18 A closed subspace of a complete uniform space is complete. PROPOSITION 4.19 A uniform space is compact if and only if it is complete and precompact.
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The proofs of the two propositions above are essentially the same as the proofs of Propositions 1.23 through 1.25 with sequences being replaced by nets. They are left as an exercise (Exercise 5). In a 1977 paper titled A note on uniform paracompactness that appeared in the Proceedings of the American Mathematical Society (Volume 62, Number 2, pp. 359-362), M. Rice introduced the concept of uniform paracompactness which is defined as the property that every open covering has a uniformly locally finite open refinement. A refinement V of a covering U is said to be uniformly locally finite if there exists a uniform covering W such that each member of W meets only finitely many members of V. THEOREM 4.6 A uniform space is cofinally complete if and only if it is uniformly paracompact. Proof: We will prove that a uniform space (X, Il) is cofinally complete if and only if each directed open covering of X is uniform and then use the equivalent fonn of unifonn paracompactness given in Exercise 7(b) to finish the argument. To prove the sufficiency. assume each directed open covering is uniform and suppose (X, Il) is not cofinally complete. Then there exists a cofinally Cauchy net 'V:D ~ X that does not cluster. For each a E D put H a = {'V(o) I 0 ~ a} and let Fa = et(H a). Then nF a = 0 since'll does not cluster. For each a E D put U a = X - Fa. Then U = {U a} is a directed open covering of X, so U E Il. But for each a E D, 'V is eventually in X - U a which is a contradiction. Therefore, (X, Il) is cofinally complete. Conversely, assume (X, Il) is cofinally complete. Let U = {U a Ia ED} be a directed open covering of X where (D, <) is a directed set. If X = U a for some a E D then U E Il so assume Fa = X - U a =f. 0 for each a. Put E = {(F a,x) Ia E D and x E Fa} and for each a let O. But Fe = {\jI(Fe, x)lx E Fo},soVcU e . Therefore, V< U,SO UE Il.-
EXERCISES 1. Prove Propositions 4.12 and 4.13. 2. Prove Proposition 4.14. 3. Let (X, Il) be a uniform space and let U E Il. A c X is said to be II-small if A c U for some U E U. A is U-Iarge if its complement is U-small. A
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collection H of proper subsets of X is heavy if for each U E ~ there is an HE H that is U-large. Show that X is complete if and only if each heavy covering has a finite subcovering. 4. A collection U of proper subsets of X is said to be bound to another collection V if for each finite subcollection W of U that does not cover X. neither does Wu {V} for each V ::j; X in V. A collection that is bound to a heavy collection is called heavily bound. Show that a uniform space is cofinally complete if and only if each heavily bound open covering has a finite subcovering. 5. Prove Propositions 4.18 and 4.19. 6. Prove Theorem 4.4. UNIFORMLY PARACOMPACT SPACES 7. [M. Rice, 1977] The following statements are equivalent: (a) X is uniformly paracompact. (b) Each directed open covering is uniform. (c) If U is an open covering of X, there exists a uniform covering V such that U I V has a finite subcovering for each V E V. 8. [M. Rice, 1977] A locally compact uniform space is uniformly paracompact if and only if it is uniformly locally compact. A uniform space (X, ~) is said to be uniformly locally compact if there exists a U E ~ consisting of compact sets. UNIFORML Y PARACOMPACT METRIC SPACES 9. [M. Rice, 1977] The collection of points of a uniformly paracompact metric space that admit no compact neighborhood is compact. 10. [A. Hohti, 1981] A necessary and sufficient condition for a metric space to be uniformly paracompact is that there exists a compact K c X with X Star(K,c:) is uniformly locally compact for each c: > O. 11. [A. Hohti, 1981] If (X, d) and (Y, 8) are uniformly paracompact metric spaces, then (X x Y, d x 8) is uniformly paracompact if and only if one of the following conditions hold: (a) either (X, d) or (Y, 8) is compact or (b) both (X, d) and (Y, 8) are locally compact. 12. [A. Hohti, 1981] Let a be an infinite ordinal. Put H(a)
= ([0,1] x a)/E Where
4.4 The Completion ofa Uniform Space
=
= = =
97
xEy if and only if x y or P I (x) 0 PI (y). Define a metric d on H(a) by d(x,y) Ip I (x) - PI (y) 1if P2(X) P2(Y) or d(x, y) P I (x) + PI (y) otherwise. Then (H(a), d) is the hedgehog metric space with a spines. H(a) is a uniformly
=
=
paracompact metric space that is not locally compact. It is therefore a counter example to a statement by P. Fletcher and W. Lindgren in 1978 that the product of a uniformly locally compact space with a C-complete (uniformly paracompact) space is C-complete.
4.4 The Completion of a Uniform Space As seen in Chapter 2 (Section 2.1, Exercise 1), every metric space is a uniform space. Our first proposition below states that a complete metric space is also complete when considered as a uniform space. This may not seem surprising at first glance, but it might if we rephrase it in the following manner: In a metric space, the convergence of all Cauchy sequences forces the convergence of all Cauchy nets (of all cardinalities). The proof of this proposition is left as an exercise (Exercise 1). PROPOSITION 4.20 A complete metric space is also complete when considered as a uniform space.
Our next result will show that Proposition 4.20 can be generalized to uniform spaces; i.e., that the convergence of all Cauchy nets on a certain ordered set forces the convergence of all Cauchy nets. We will then use these specialized nets to construct a completion of the uniform space from equiValence classes of these specialized nets in a manner similar to the construction of the metric completion in Chapter 1. Our first step will be to define these specialized nets. For this let (X, J.l) be a uniform space and let v be a cofinal subset of J.l of least cardinality. Then v is a basis for J.l. Next, well order v by some ordering < such that the cardinality of each initial interval is less than the cardinality of v. From the proof of the Ordering Lemma, it can be seen that there exists a cofinal subset A of v (with respect to the ordering <*) such that the well ordering < is compatible with <* on A (i.e., U <* V implies that V < U). Then A is a well ordered basis for J.l of least cardinality such that the well ordering < is compatible with the directed ordering <* and the cardinality of each initial interval (with respect to <) is less than the cardinality of A. We call A a fundamental basis for J.l and a net {xa 1 a E A} C X a fundamental net with respect to A. Let {x a Ia ED} be a net in X and let (E, <) be a directed set. A function ~:E ~ D is called a compatible function if whenever a, ~ E E with a < ~ then ~(~) is not strictly less than ~(a) in D. A net {y 13 ~ E E} c X where y 13 = x ~(13) for each ~ E E is said to be contained in the net {x a}. Note that a cofinal function is a compatible function so a subnet of a net is contained in the 1
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net. Also note that a compatible function need not be cofinal so that a net {y j3 } contained in a net Ix a} need not span the net {x a} as a subnet would.
THEOREM 4.7 Each Cauchy net 4> contains afundamental Cauchy net \jf such that 4> converges to a point p if and only if\jf converges to p. Proof: (Construction of \jf) Let 4> = {x a 1a ED} where D is ordered by =:;. Well order D by < such that the cardinality of each initial interval is less than the cardinality of D. Then there is a cofinal E c D with respect to =:; such that < is compatible with =:; on E. Now 4>£ = {x a 1a E E} is a Cauchy subnet of 4>. For each a E A there is a Va E a and a residual Ra c E with respect to =:; such that 4>£(R a) C Va' Put Va = Star{Ua. a) and let U = {Va} and V = {Va}. Then U has the finite intersection property and for each Va E U. if V is another member of a with 4>£ eventually in V then V c Va' Also, V is directed by set inclusion; i.e.,ifa,bE AthereisacE AwithVccVanVb • ForeachaE AdefineEa={D E E 14>£(8) E Va} and for each b E A let ~(b) be the first element of Eb (with respect to <). Then ~:A ~ E is a compatible function and \jf = 4>£ © ~ is a fundamental net contained in 4>. Let a E A and pick bE A with b <* a. Then Vb c W for some WE a so \jf(b) = Xt;(b) E W. Also, if c E A such that c <* b then Vc c Vb' Thus \jf(c) E W. Put R = {c E AI c <* a}. Then R is residual in A and \jf(R) c W so \jf is Cauchy. (Proof 4> converges if \jf converges) Assume \jf converges to p E X and N is a neighborhood of p. Then there are a,b E A with a <* band Star(p,b) eN. Also, there is aCE A with c <* a and \jf(c) E Star(p,a). Then p and \jf(c) both belong to some Wa E a. Also \jf(c) E Vc and Vc C Va' Hence Star(p,a) c Star(Wa , a) c Wb for some Wb E b and Va C Star(Wa, a) c Wb. Now 4>(Ec) = 4>£(Ec) c Vc C Va so 4>(RJ c Va' Since Rc is residual in E with respect to =:;, it is cofinal in D with respect to =:;, so 4> is frequently in Va C Wb c Star(p,b) eN. Since 4> is Cauchy, 4> converges to p. (Proof \jf converges if 4> converges) Assume 4> converges to p and N is a neighborhood of p. Then there are a,b E A with a <* band Star(p,b) c N. Also, there is a residual ReD with respect to =:; with 4>(R) c Wa for some Wa E a such that p E Wa' Then RnE is residual in E and 4>£(RnE) c Wa' Since Ra is residual in E so is RnR a. Thus there is a 8 E RnRa with Xc = 4>£(8) E Wa' But Xc E Va, so WanVa i= 0. Also, \jf(a) = Xt;(a) implies \jf(a) E 4>£(Ea) eVa. Moreover, if C E A with c <* a then Vc C Va so \jf(C) E Va' Thus \jf is eventually in Va. Now a <* b implies there is a Wb E b with WauVa C Wb so P E Wb which implies Va C Star(p,b) eN. Hence \jf converges to p.Until now, we have avoided the definition of a subspace of a uniform space. Subspaces are one of the fundamental constructions to be dealt with in the next chapter. The approach so far has been to avoid these constructions (i.e., building new uniform spaces from old ones) as long as possible in order to examine the behavior of the generalizations of sequences and convergence
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(from metric spaces) and some of their topological consequences in uniform spaces as soon as possible. Strictly speaking, the completion of a uniform space can be introduced without introducing the concept of a subspace, but the most useful way of thinking of the completion is, that it is a (perhaps) larger uniform space that contains the original uniform space as a subspace. Consequently, we now give a simple definition of a uniform subspace, but will revisit the concept in the next chapter in a more formal setting. A subset A of a uniform space (X, 11) can be given a uniform structure that is derived from the uniformity 11 in the following way: for each U E 11 put UA = {UnAIU E U}. Then let IlA = {U A IU E Ill. It is easily shown that IlA is a uniformity on A. IlA is called the uniformity induced on A by the uniformity 11. IlA is also said to be the uniformity of 11 relativized to A. It should be noted that the uniformity IlA causes the identity mapping iA:A ~ X to be uniformly continuous. Our next task is to construct a completion for (X, 11) from the fundamental Cauchy nets. For this let L be the set of all fundamental Cauchy nets in X and define an equivalence relation - on L by - \jf if for each U E A there is a U E U such that and \jf are both eventually in U. Let X' be the set of all equivalence classes with respect to - and for each U E 11 put U~ = {U~ IU E U} where U~
= {' E x'i is eventually in U for each E '}.
Then Il~ = {U~ I U Ell} is a basis for a uniformity for X'. Let 11' be the uniformity generated by Il~. Then (X', 11) is a uniform space. Note that if x E X there is a unique <1>' E X' that consists of all fundamental Cauchy nets that converge to x. For each x E X let i(x) denote the unique equivalence class <1>' whose members converge to x. Then i:X ~ X' is well defined and since X is Hausdorff, i is one-to-one. Let U Eiland U E U. It is easily shown that i(U) = U~ni(X). Define i(U) = Ii(U) I U E U} for each U E 11. Then for each U' E 11' there is a U~ E Il~ that refines U' so U~ni(X) = {U~ni(X) I U E U} = i(U) refines U'ni(X). But then U refines j-l(Uni(X)) so i-1(U'ni(X)) E 11. Consequently i(X) is a uniform subspace. It is easily shown that i(X) is dense in X'. To show X' is Hausdorff let <1>' and \jf' be distinct elements of X'. Then there exists E <1>' and \jf E \jf' such that is not equivalent to \jf. Hence there is a U E 11 such that either or \jf is not eventually in U for each U E U. Note this implies that if VEil with V < U then either or \jf is not eventually in V for each V E V. Pick WEll with W** < U. Since both and \jf are Cauchy there are WI, W 2 E W with eventually in WI and \jf eventually in W 2. Now W* < U implies W InW 2 = 0 so W;nW; = 0.
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Let 8 E q>'. There is aWE Wand residual P,Q C A with 8(P), q>(Q) c W. Also, there is a residual RCA with q>(R) C WI' Then S = PnQnR is residual in A, 8(S) and q>(S) c Wand q>(S) c WI, so there is a V I E W* with W c V I. Thus q>' E V;. Similarly, there is a V 2 E W* with 'Jf' E V;. W** < U implies V I n V 2 = 0 so V; n V; = 0. Since V; and V; are neighborhoods of q>' and 'Jf' respectively in X', X' is Hausdorff. It remains to show (X', fJ.') is complete. Let {q>~ IU E A} be a fundamental net in i(X). Then for each U E A, there is an Xu E X with q>~ = i(xu)' Put 'Jf(U) =Xu' Then 'Jf:A ~ X is a fundamental net. It is easily seen that 'Jf is Cauchy. Let U' be an open set in X' containing 'Jf', Then there is a UAE fJ.A with 'Jf' E Star('Jf', U c U' so there is a VA E U' with 'Jf' E VA C U'. Then 'Jf = {Xu} is eventually in U so q>~ E U'. Thus {q>~ I converges to 'Jf'. A
)
Next let {q>~ IU E AI be a fundamental Cauchy net in X'. For each a E A there is a residual Ra c A with {q>~ I U E Ra) c u~ for some u~ E aA. Let V~ = Star(U~, a A) and put U' = {U~} and V = {V~}. Just as in the proof of the preceding theorem, for each a E A, if UAE a and {q>~} is eventually in U A, then UAnU~ "F 0 so U AC V~, and if a,b E A with a <* b then V~ c V~. A
Since i(X) is dense in X', for each a E A there is an Xa E X with i(xa ) EVa' Then {i(x a )) is a fundamental net in i(X). It is easily shown that {i(xa)} is Cauchy and consequently converges to some p' E X'. Let N be a neighborhood of p' and b E A with Star(p', b A) c N. Pick a E A with a <* b. Since {i(xa)} converges to p' there is aCE A with c <* a and i(xJ E Star(p', a A). Thus p' and i(xJ both belong to some W A E aA. Now i(xc) E V; and {q>~} is eventually in U; c V;. Hence there is a W~ E a A with V; c W~ so W~nU~ "F 0. Since i(xc ) E WAnW~ we have W A c Star(W~, a A) c W~ for some W~ E bA. Since U~ c W~, {q>~} is eventually in W~ c Star(p', bA) c N and we conclude {<1>~} converges to p'. Then by the above theorem, (X', fJ.') is complete. (X', fJ.) is called the completion of (X, fJ.). Since the function i is one-toone and uniformly continuous we often identify X with the dense uniform subspace i(X) of X'. In this case, i(x) = X for each X E X; i.e., the function i is the identity mapping on X. Using this identification, we notice that for each U E fJ. there is a U' E fJ.' such that U = i-I (U'), so U = U' nX = {U'nX I U' E U'}. Also, if V' E fJ.' then V' nX E fJ.. We record these results as: THEOREM 4.8 Every uniform space has a completion; i.e., ij(X, fJ.) is a uniform space, there is a complete uniform space (X', fJ.') and a one-to-one uniformly continuous function i:X ~ X' such that i(X) is a dense uniform subspace of X',
Theorem 4.8 corresponds to Theorem 1.15 for metric spaces. Theorems 1.16 and l.17 can also be generalized to uniform spaces as follows:
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THEOREM 4.9 iff is a uniformly continuous function on a subset A of a uniform space X into a complete uniform space Y then f has a unique uniformly continuous extensionj' to CI(A). ProoF Let!l be the uniformity on X and v the uniformity on Y. For each x A let {x a} be the constant fundamental Cauchy net (x a = x for each ex) that converges to x and for each x E CI(A) - A pick a fundamental Cauchy net {x a } c A that converges to x. Let V E v. Since f is uniformly continuous, there exists a U E 11 such that whenever a,b E A with a,b E U for some U E U, then f(a),f(b) E V for some V E V. But then the net {f(x a )} is Cauchy in Y for each x E CI(A). Since Y is complete, {f(x a )} converges to some x' E Y. Define j':CI(A) -7 Y by j'(x) = x' for each x E CI(A). Clearly,j'(a) = f(a) for each a E A so j' is and extension of f. E
To show j' is well defined, let x E CI(A) and let {Ya} be another fundamental Cauchy net converging to x. Suppose x' -j:. y'. Then there exists a V E v with S(x',v)nS(y',v) = 0. By the uniform continuity of f, there exists a U E 11 such that a,b E U E U implies f(a), feb) E V for some V E V. Since both {x a} and {Ya} converge to x, there exists a ~ such that ex > ~ implies x a,Ya E U for some U E U which in turn implies f(x a), f(Ya) E V for some V E V. But this is impossible since {f(x a)} is eventually in S(x',V) and {f(Ya)} is eventually in S(y',v). Hence x' = y' so j' is well defined. To show j' is uniformly continuous on CI(A), let W E v and pick V E v with V* < W. Since f is uniformly continuous on A, there exists a U E 11 such that whenever a,b E A with a,b E U for some U E U, then f(a),f(b) E V for some V E V. Let Z E 11 with Z* < U. Let X,Y E CI(A) such that X,Y E Z for some Z E Z. Since {xa} converges to x, {Yu} converges to Y, {f(x a )} converges to x', and {fey a)} converges to y', it is possible to choose a ~ such that ex > ~ implies x a,x E Z I, and YaS E Z2 for some Z j, Z2 E Z, and x',f(x a ) E VI and y',f(ya) E V2 for some Vj, V2 E V. Then Xa,Ya E Star (Z,Z) c U for some U E U. Hence f(x a), f(Ya) E V for some V E V. But then x',y' E Star(V,V) c W for some WE W. Therefore,x,y E Z which impliesj'(x),j'(y) E W, so j' is uniformly continuous. To show j' is unique, suppose F is another uniformly continuous extension of ffrom A to CI(A). Let x E CI(A) - A. Then {x a } converges to x. Since both j' and F are continuous, it follows that {j'(x a )} converges to j'(x) and {F(xa)} converges to F(x). But {xa} c A implies {j'(x a )) = {f(x u )} = {F(xa)}' Since a net in a Hausdorff space has (at most) a unique limit, j'(x) = F(x), so j' is unique. THEOREM 4./0 For each uniform space (X, 11), its completion (X', 11') is unique; i.e., if (X', 11') is another completion of (X, 11) then there is a uniform homeomorphism h:X' -7 X' that keeps each point of X fixed.
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Proof: Let j:X ~ X' denote the uniform homeomorphism that maps X into the completion X'. By Theorem 4.9 there exists a unique uniformly continuous extension }':X' ~ X'. Since}' is an extension of j we have J = © i where i:X
r
~
X' denotes the uniform homeomorphism that maps X into its completion X'. Similarly, there is a unique uniformly continuous extension t:X' ~ X" of i. Then i = (© j. For each x' E X' put g'(x') = i'(j'(x')). Then g' = }' © ( so g':X' ~ X' is uniformly continuous. If x E X then g'(x) = {(j'(x» = {(j (x» = i (x) = x so g' is an extension of the identity map i:X ~ X'. But by Theorem 4.9, this extension is unique so g' = i' (the identity map on X'). Hence}' © { = i' so { = (j'r i . But then X' and X' are uniformly homeomorphic. -
EXERCISES 1. Prove Proposition 4.20. 2. Show that the completion of a unifonn space is compact if and only if the uniformity is precompacr. 3. Show that a complete subspace of a Hausdorff uniform space is closed. 4. [K. Morita, 1951] Let 11* be the uniformity of!J.X. For each U = {U a I in 11, put U' = {U~ I where U~ = !J.X - Clf1X(X - U a) for each a. Then {U'I U E III is a basis for 11*. CAUCHY FILTERS AND WEAKLY CAUCHY FILTERS A filter F in a uniform space (X, 11) is said to be Cauchy if for each U E 11, there exists an F E F such that FeU for some U E U. It is said to be weakly Cauchy if for each U E 11, there exists aU E U with UnF of- 0 for each F E F. 5. Show that (X, 11) is complete if and only if each Cauchy filter in X converges. 6. [N. Howes, 1971] (X, 11) is cofinally complete if and only if each weakly Cauchy filter in (X, u) clusters. 7. [H. Corson, 1958] X is paracompact if and only if each weakly Cauchy filter in (X, u) clusters.
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4.5 The Cofinal Completion or Uniform Paracompactification Since each uniform space has a unique completion. it is natural to define a unifonn space (X'. f..l') to be a cofinal completion of the uniform space (X. f..l) if (X'. f..l') is cofinally complete and (X. f..l) is uniformly homeomorphic to a dense uniform subspace of (X'. f..l') and to ask: "When does (X. f..l) have a cofinal completion. and if a cofinal completion exists. is it unique?" Providing answer~ to these questions is the objective of this section.
It turns out these same techniques can be used to provide answers to a variety of other questions asked by K. Morita and H. Tamano. In the late fifties. Tamano considered the following question: "What is a necessary and sufficient condition for a uniform space to have a paracompact completion?" Although he did not arrive at a solution. he obtained elegant characterizations of completeness. paracompactness and the structure of the completion by means of a concept called the radical of a uniform space. These results were published in the Journal of the Mathematical Society ofJapan in 1960 (Volume 12, No.1, pp. 104-117) under the title Some Properties of the Stone-Cech Compactijication. We will analyze these results in Chapter 6. In 1970, K. Morita presented a paper titled Topological completions and M-spaces at an international Topology Conference held at the University of Pittsburgh. In Section 7 of that paper, five unsolved problems were listed including a special case of Tamano's question mentioned above and the question: "What is a necessary and sufficient condition for a Tychonoff space to have a LindelOf topological completion?" By a topological completion we mean the completion with respect to the finest uniformity. In this section and in later chapters, we provide answers to all these questions. All of the solutions are in terms of the cofinally Cauchy nets and their behavior with respect to various uniformities. Since some of these questions ask if the completion of a unifonn space has a certain topological property, one might be interested in knowing if there are solutions in terms of topological properties, or if topological properties exist that are only necessary or only sufficiem. Since the literature is rich with characterizations of paracompactness and the LindeWf property, one might expect a variety of solutions to these problems. However, at the present time, this area is largely unexplored. We define a uniform space to be preparacompact if each cofinally Cauchy net has a Cauchy subnet. Recall that a uniform space is precompact if every net has a Cauchy subnet. Consequently, preparacompactness is a generalization of precompactness. To continue the parallel. recall that complete precompact spaces are compact and the completion of a precompact space is compact. We will see that complete preparacompact spaces are paracompact and that the completion of a preparacompact space is paracompact.
104
4. Completeness, Cofinal Completeness and Uniform Compactness
A unifonn space will be called countably bounded if each unifonn covering has a countable subcovering. If (X, Jl) is a unifonn space and (X*, Jl*) its completion, let u* denote the universal unifonnity for X* and let v be the uniformity induced on X by u*. The v unifonnity is called the uniformity derived from u* or simply the derived uniformity. A directed set D is said to be co directed or countably directed if for each countable {d;} c D there is a d E D such that d; ~ d for each i. A net {xa I a E D} is co directed if D is an co directed set. LEMMA 4.1 A completely regular space is LindelOf if and only if each co directed net clusters. Proof: Assume that each co directed net clusters in the completely regular space X. Suppose X is not Linde16f. Then there is a covering U of X having no countable sUbcovering. For each countable V c U put G(V) = u {u IU E V} and F(V) = X - G(V). Let S be the set of all countable subsets of U and for each V E S pick Xv E F(V). The assignment V -7 Xv defines an co directed net {xv IV E S} in X that clusters to some p E X where S is directed by set inclusion. Let U E U such that p E U. Then {xv} is eventually in F( {U}) and hence cannot be frequently in U which is a contradiction. Consequently X is Linde16f.
Conversely assume X is LindeWf and let {x a Ia ED} be an co directed net in X. Let U be a member of the e unifonnity of X. Then there is a countable subcovering V = {Vd such that VEe and V refines U. Suppose {x a} is not frequently in some member of V and put D; = {O E D Ix Ii E V;}. Then there exists a 0; E D such that 0 ~ 0; for each 0 E D; or else {Xli 10 E D;} would be frequently in V;. Since {x a} is CO directed there is a 0 E D such that 0; ~ 0 for each i. Now Xli E Vj for some j since V covers X. Hence 0 E D j which implies 0< OJ. But OJ ~ 0 which is a contradiction. Therefore Xa must frequently be in some member of V. Since V was chosen arbitrarily, {xa} is cofinally Cauchy with respect to e and therefore clusters by Theorem 4.4. • The following theorem appeared in an article titled On completeness in the Pacific Journal of Mathematics in 1971 (Volume 38, pp. 431-440). THEOREM 4.11 (N. Howes, 1970) Let (X, Jl) be a uniform space and v the derived uniformity. Then: (1) (X, Jl) has a paracompact completion if and only if (X,v) is preparacompact and (2) (X, Jl) has a LindelOf completion if and only if (X,v) is countably bounded and preparacompact, and (3) (X, Jl) has a compact completion if and only if (X,v) is precompact. Proof of (1): Let (X', Jl') be the completion of (X, Jl) and let u' be the universal unifonnity for X'. Then (X,v) is a dense unifonn subspace of (X', U). Assume (X,v) is preparacompact and that ",:D -7 X' is a cofinally Cauchy net with
4.5 The Cofinal Completion or Uniform Paracompactification
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respect to u'. Since (X', Il') is complete, so is (X', u'). Let E = D x u' and define on E by (d, U') $ (e,v') if d $ e and V' <* U'. For each (d, U') E E put S(d,U,) = a for some a E X such that a and 'V(d) both belong to some U' E U'. Then the correspondence (d, U') ---+ Sed, U') defines a net S:E ---+ X. $
Let U' E u' and pick V' E u' with V' <* U'. Since 'V is cofinally Cauchy there is a cofinal C c D with 'V(C) c V' for some V' E V'. Put A = {(d,W')ldE C and W'<* V'}. Then A is cofinal in E. Let (d,W') E A. Then S(d,W') = Y E X such that y and 'V(d) both belong to some W' E W'. Since (d,W') E A, dEC which puts 'V(d) in V'. Consequently we have:
YE Star(y',W') c Star(Y',V') c U' for some U' E U'. Therefore SeA) c U' which implies S is cofinally Cauchy in (X', u'). But SeE) c X implies S is cofinally Cauchy in (X,v). Consequently S has a Cauchy sub net ~. But then ~ converges to some x' E X'. Therefore S clusters to x'. It remains to show that 'V clusters to x'. For this let 0 be an open set containing x'. Then there is a U' E u' such that x' E Star (x',U') cO where the members of U' are open sets. Pick V' E u' such that V' <* U'. Let S be cofinal in E such that S(S) c V' for some V' E V' containing x'. Put D(S)
=
{d E DI (d,W') E S for some W' <* V'}.
Then D(S) is cofinal in D. For each d E D(S), 'V(d) and S(d,W') are contained in some W' E W' for some (d,W,) E S where W' <* V' which implies 'V(d) and S(d,W,) are contained in some Y'j E V' for each (d,W') E S. But (d,W') E S implies S(d,W') E V'. Hence 'V(d)
E
V'uY~ c Star(Y',V') c U'
for some U' E U' . But x' E V' implies x' E U/. Then U' c 0 since Star (x',U') c O. Consequently 'V(d) E 0 for each dE D(S) which implies 'V clusters to x'. Therefore each cofinally Cauchy net in X' clusters which implies (X', u') is cofinaIly complete. But then X' is paracompact by Theorem 4.4. Conversely suppose X' is paracompact. By Theorem 4.4 (X', u') is cofinally complete. Let 'V be a cofinally Cauchy net with respect to v. Since (X,v) is a uniform subspace of (X', u,), we know that 'V is cofinaIly Cauchy in (X', u'). Also since (X', u') is cofinally complete, \jI clusters to some p EX'. But then 'V has a subnet e that converges to p. Then e is Cauchy in (X', u'). But SeX and therefore e is Cauchy in (X,v). Consequently, each cofinaIly Cauchy net in (X,v) has a Cauchy subnet so that (X,v) is preparacompact.
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4. Completeness, Cofinal Completeness and Uniform Compactness
Proof of (2): Assume first that X' is LindelOf. Then X' is paracompact and hence (X', u') is cofinally complete. But then (X,v) is preparacompact as was shown in part (1). Next let V E v. Then V has a uniform refinement U consisting of closed sets. For each U E U put U' = Clx'(U) and let U' = {U' Iu' E U). Then U E u' and hence has a countable subcovering say {V;). Then { V,) covers X. In fact, if P E X then P E X' which implies P E vj for some positive integer j. Hence P E Clx'(Uj)' Let 0 be open in X such that P E O. Then 0 = 0' nX for some 0' that is open in X'. Now P E 0' which implies 0' nVj :j. 0 since P E Clx'(Uj)' Then there is atE 0'nVj. t E Vj implies t E X so we have (O'nX)nVj :j. 0. Hence OnVj :j. 0 so that P E Clx(Uj) = V j . Since {V j ) covers X there exists some {Vj ) C V such that uVj = X. Consequently X is countably bounded.
Conversely assume (X,v) is countably bounded and preparacompact. (X',j.!') is paracompact by part (1) so that (X', u') is coftnally complete by Theorem 4.4. Let U' E u'. Since X' is paracompact there exists a locally finite open refinement ~' = {Vp IBE B). Since X is norrn~l we c~n shrink V' to an open covering W = {W~ IB E B} such that Clx'(W~) C V~ for each BE B. Then W' is also locally finite. Since W' is an open covering in a paracompact space X', it must be a member of the universal uniformity u'. But then W = {W'nX IW' E W'} belongs to v which implies there is a countable subcovering {W~,) c W. Since W' is locally finite in X', {W~,) is locally finite in X'. Hence Ui'=l Clx'(W~) = CIX,(Ui'=1 W~) = Clx'(X) = X' since X is dense in X'. Therefore {Clx'(W~)} covers X and
for each positive integer i. Hence {V~,} covers X'. But since V' refines U' there exists a countable subcovering {V~,) c U' that covers X'. Therefore (X',u') is also countably bounded. We are now in a position to show that X' is LindelOf. We will use the fact that (X', u') is cofinally complete and countably bounded to show that each co-directed net in X' clusters. We then invoke Lemma 4.1 to obtain the desired result. Let \jf:D ~ X' be an co directed net and let U' E u'. Then U' has a countable subcovering say {V;}. Put D j = {d E D I \jf(d) E V;) for each i and suppose D j is not cofinal in D for each i. Then there exists a d j E D for each i such that d ~ d j for each dE D j • Since \jf is co directed there is a do E D such that d j ~ do for each i. Since {V;} covers X', \jf(d o) E vj for some positive integer j which implies do E D j which in turn implies D j is cofinal in D since D = ui'=lD j • Therefore \jf is frequently in V j • Hence \jf is cofinally Cauchy and consequently must cluster. Thus X' is LindelOf by Lemma 4.1. Proof of (3): By Problem 3 of Section 4.4, (X, j.!) has a compact completion (X', j.!') if and only if j.! is precompact. But if v is precompact then j.! c v implies j.! is precompact which in turn implies X' is compact. Conversely, if
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X' is compact then /-l' = u' which implies /-l = v and /-l is precompact which implies v is precompact. -
COROLLARY 4.2 Let u be the universal uniformity for a completely regular T I space X. Then: ( 1) (X, u) has a paracompact completion if and only if it is preparacompact, (2) (X, u) has a Linde16f completion if and only if it is countably bounded and preparacompact. COROLLARY 4.3 The completion of a preparacompact uniform space is cofinally complete. COROLLARY 4.4 A countably bounded cofinally complete uniform space is Linde16j. COROLLARY 4.5 A paracompact space is Linde16f if and only if it is countably bounded with respect to the universal uniformity.
In view of the existence of a unique completion for each uniform space, it is natural to ask when a uniform space has a cofinal completion, and if a cofinal completion exists, when is it unique? THEOREM 4.12. A uniform space has a cofinal completion if and only if it is preparacompact. In this case it is unique and identical to the ordinary completion (i.e., the ordinary completion is cofinally complete). Proof: (Sufficiency) Suppose (X, /-l) is preparacompact and (X', /-l') is its completion. It needs to be shown that (X', /-l') is cofinally complete. In the sufficiency part of the proof of Theorem 4.11.(1), it was shown that if v is the uniformity derived from /-l rather than the uniformity /-l and if (X,v) is preparacompact then (X', /-l') is cofinally complete. The proof that (X', /-l) is cofinally complete based on the assumption that (X, /-l) is preparacompact is similar so it will not be included here. (Necessity) Assume (X, /-l) has a cofinal completion (X', /-l'). Let Ix a I a E D) be a cofinally Cauchy net in (X, /-l). Then {xa} clusters to some p E X' so {xa} has a subnet {y~} that converges to p. But then {y~} is Cauchy in (X',/-l'). Since {Y/3} lies in X it is Cauchy in (X, /-l). Consequently, every cofinally Cauchy net in (X, /-l) has a Cauchy subnet so (X, /-l) is preparacompact. Since (X', /-l') is cofinally complete it is also complete. But then (X', /-l') is a completion of (X, /-l). Since the completion of a uniform space is unique this means the cofinal completion (when it exists) is identical to the completion.-
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EXERCISES 1. Show that a completely regular T 1 space is paracompact if and only if it is complete and preparacompact with respect to the universal uniformity.
2. Show that a uniform space (X, J.!) is countably bounded if and only if each directed net is cofinally Cauchy.
(t)
3. Let J.! be a uniformity that generates the topology of X and let U be an open covering of X. We call P E X a residue point for U with respect to V E J.! if there does not exist aU E U with Star (p, V) c U. Let H v be the set of residue points with respect to V. Put Fv = CI(H v ) and G v = X - Fv· Then W = {G v I V E J.!} is called the residue covering derived from U. Show that W is an open covering of X. 4. Show that a completely regular T 1 space is paracompact if and only if each residue open covering with respect to the universal uniformity has a finite subcovering. 5. Show that a completely regular T 1 space is paracompact if and only if each heavily bound open covering (Exercise 5, Section 4.3) with respect to the universal uniformity has a finite subcovering. 6. Show that in a completely regular T I space the following are equivalent: (a) the LindelOf property, (b) each residue open covering with respect to e has a finite subcovering, (c) each heavily bound open covering with respect to e has a finite subcovering. 7. A topological space is said to be entirely normal if the collection of all neighborhoods of the diagonal in X x X forms an entourage uniformity (Exercise 4, Section 2.1). Show that entire normality and almost-2-fully normal (Exercise 6, Section 3.2) are equivalent. 8. A net in a topological space X will be called cofinally t1 Cauchy if for each open covering U of X there is apE X such that the net is frequently in S(p, U). X is cofinally t1 complete if each cofinally t1 Cauchy net clusters. Show that an entirely normal space is paracompact if and only if it is cofinally t1 complete. 9. Show that a metacompact space is cofinally t1 complete. 10. Show that entire normality implies collection wise normality.
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11. Show that a metacompact space is paracompact if and only if it is collection wise nonnal. 12. RESEARCH PROBLEM We would like to have a characterization of preparacompactness in tenns of coverings, perhaps analogous to the characterization of precompactness in tenns of coverings. The following result is in this direction, but something better is needed. A collection ~ of subsets of X is said to be a directed collection if for each A,B E ~, AuB E ~.
THEOREM A uniform space (X, !!) is preparacompact if and only if each heavily bound collection having no finite subcollection that covers X is contained in a heavy directed collection. See Exercises 3 and 4 of Section 4.3.
Chapter 5
FUNDAMENTAL CONSTRUCTIONS
5.1 Introduction In this chapter we consider some important constructions of uniform spaces from other uniform spaces. Our first concern will be to consider the so called classical constructions that are studied for most spaces and algebraic structures that arise in the study of mathematics, namely subspaces, sums, products and quotients. Our approach will be to derive these constructions from a few fundamental concepts. These fundamental concepts take the form of limits of collections of uniformities. We will make these concepts precise in the next section. The reason for waiting until now to introduce the fundamental constructions is that many of the interesting results about these constructions involve concepts related to completeness. For example, it can be shown that the product of complete uniform spaces is a complete uniform space. Without these other concepts, it is difficult to appreciate the utility of these constructions. We opted instead to first develop the theory of completeness in uniform spaces and then see how it applies to these constructions. After the classical constructions, we proceed to some other constructions (some of which also apply to a variety of other structures such as topological spaces). We should remark that we have already seen one such construction; namely, the completion of a uniform space. Of great interest to us will be the concepts of the hyperspace of a uniform space, the inverse limit of a directed family of uniform spaces, the weak completion of a uniform space and the spectrum of a uniform space. If the hyperspace is complete, the original space is said to be supercomplete. Like cofinal completeness. supercompleteness turns out to be a strong form of completeness. In fact, it was recently discovered that supercompleteness is a property that lies between completeness and cofinal completeness. This result will allow us to strengthen several results from Chapter 4. The concept of supercompleteness was introduced by S. Ginsburg and J. Isbell in 1954 in an abstract titled Rings of convergent functions that appeared in the Bulletin of the American Mathematical Society, Volume 60, page 259. But the definition of supercompleteness in the simple form mentioned above is
5.2 Limit Uniformities
III
due to Isbell in a paper that appeared in the Pacific Journal of Mathematics in 1962 titled Supercomplete spaces (Volume 12, pp. 287-290). The concept of the inverse limit of a family of uniform spaces leads to the concept of the spectrum of a uniform space, and the spectral analysis of uniform spaces whose spectra exist. The study of uniform spaces by means of their spectra has been pursued by B. Pasynkov (1963) and K. Morita (1970). Also, the concept of the inverse limit of a directed family of uniform spaces is needed to express the weak completion of a space with respect to a uniformity introduced by K. Morita in 1970. Finally, we will introduce the locally fine corefiection and the sUbfme corefiection of a given uniform space. These were introduced by Ginsberg and Isbell in 1959 and shown to be equivalent by J. Pelant in 1987. This new construction has a variety of uses. In the last section of this chapter we will introduce the concepts of categories and functors. This material is optional as far as being needed in later chapters. It is included for two reasons. First, much of the literature of uniform spaces after Isbell makes use of the category theoretic vocabulary, and second, category theory, like set theory, is a tool that is often helpful in the study of uniform spaces. The term "coreflection" refers to a category theoretic concept, and it will be shown that the locally fine coreflection satisfies this notion. Thereafter, categories and functors will only be used in the exercises.
5.2 Limit Uniformities Given a collection {'ta } of topologies for a set X, there exists a finest topology that is coarser than each 'ta called the infimum topology on X with respect to {'ta } and denoted infa't a . Similarly, there exists a coarsest topology that is finer than each 'ta called the supremum topology on X with respect to {'t a } and denoted supa't a . To see this, note that n'ta is a topology for X that is coarser than each 'tao If't is another topology for X that is coarser than each 'ta , then 't c n'ta so n'ta is the finest topology for X that is coarser than each 'ta (i.e., infa't a = n'ta)' Also note that if L is the collection of topologies that are finer than each 'ta , then L 7= 0 since the discrete topology belongs to L. Consequently, nL is a topology for X. For each a, 'ta C cr for each cr E L. Hence'ta c nL for each a so nL is finer than each 'tao If't is another topology finer than each 'ta , then 't E L which implies nL C 'to Therefore, nL is the coarsest topology that is finer than each 'ta (i.e., sUPa't a = nL). It can be shown (Exercise 1) that a sub-base for sUPa't a is the set u't a . The concepts of infimum and supremum uniformities are analogous to the topological concepts. If lila} is a collection of uniformities for a set X, there is a finest uniformity that is coarser than each Ila called the infimum uniformity and denoted infa/la. There also exists a coarsest uniformity that is finer than each Ila called the supremum uniformity and denoted supalla. By Theorem
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112
2.1, ul-4x is a sub-basis for a uniformity !-lon X which implies I ni'=1 Ua,
IUa,
E
1-4x, for some i = 1 ... n} is a basis for!-l. If!-l' is another uniformity for X that is finer than each I-4x then !-l' must contain each ni'=1 Ua, so !-l C !-l'. Hence!-l = supa!-la. Similarly, if L = {O'y I'Y E G} is the collection of all uniformities that are coarser than each !-la then L :F 0 since I X} E L. Put v = sUPyO'y. If V E v then by the definition of v, there exists 0'1 .. . O'n ELand Wi E O'i for each i = 1 ... n such that ni'=1 Wi < V. For each ex, O'i C I-4x for each i = 1 ... n so Wi E I-4x for each i = 1 ... n which implies ni'=1 Wi E !-la. Hence V E I-4x so !-l C !-la for each ex. So VEL and by the definition of v, v is finer than any other member of L so v is the finest uniformity that is coarser than each !-la. Therefore v = infal-4x.
PROPOSITION 5.1 If l!-la} is a collection of uniformities for a set X and I 'ta } is the corresponding collection of topologies generated by the !-la's. then the topology generated by Supal-4x is Supa't a . Proof: Let 't be the topology generated by supa!-la. Since supa!-la is finer than each !-la, 't is finer than each 'ta so sUPa't a C 'to If U E 't then for each P E U there exists a basic covering n7=1 Ua, E Supa!-la such that S(p, n7=1 Ua) C U. Now S(P,U a) E sUPa'ta for each i = 1 ... n since sUPa't a is finer than 'ta, for each i = 1 ... fl. But then ni'=1 S(P,U a,) E supa't a . If x E ni'=1 S(P,U a,), then X,P E U a, for some U a, E Ua, for each i = 1 ... n. But then x,P E ni'=1 U a, E ni'=1 Ua, which implies x E S(p, ni'=1 U a,). Therefore
which implies U E supa't a . Hence't
C
sUPa't a which implies 't = sUPa't a .-
Unfortunately, a proposition similar to 5.1 does not hold for infa!-la. Whereas u!-la and u'ta are sub-bases for the uniformity supa!-la and the topology sUPa't a respectively, n!-la is not necessarily even a uniformity while n'ta = infa'ta . The reason for this is explored in the exercises at the end of the section. Letf:X ~ Y be a function from a set X into a topological space Y. Thenf determines a coarsest topology 't/ on X such thatfis continuous. A basis for the open sets in 't/ is the collection B = {f-I (U) I U is open in Y}. If F = {fa:X ~ Y a Iex E A} is a family of functions from X into topological spaces Y a , we define the topology 'tF to be suPa't/rJ.' 'tp is called the projective limit topology for the collection F. It can be shown (Exercise 3) that a sub-basis for 'tF is the set S = lral (U a) I U a is open in Y a for some ex} and that 'tp is the coarsest topology on X such that each fa in F is continuous. Similarly, if g:Y ~ X is a function from a topological space Y onto a set X, there is a finest topology 't g on X such that g is continuoUS. A basis for the
5.2 Limit Uniformities
113
open sets in 't g is the collection B = (U c Xlg-I(U) is open in Yl. If G = {g a: Y a ~ X I ex E A} is a family of functions from topological spaces Ya onto X, we define the topology 'tG to be in!a't gex • 'tG is called the inductive limit topology for the collection G. It can be shown (Exercise 4) that a basis for 'tG is the set B = {U c X Ig"ixl (U) is open in Ya for each ex E A} and that 'tG is the finest topology on X such that each g a EGis continuous. The concepts of projective and inductive limits are also relevant to collections of uniformities. The only difference is that now we are interested in making collections of functions uniformly continuous rather than simply continuous. If !:X ~ Y is a function from a set X into a uniform space (Y, v), then by Exercise 2 of Section 2.2, ! determines a coarsest uniformity f..lf on X such that! is uniformly continuous. A basis for f..lf is l (v) = If-I (V) IV E v}. Let F = {!a:X ~ (Ya,v a )I ex E A} be a collection of functions from X into uniform spaces (Ya,v a ). Define f..lF to be sUPaf..lfex' f..lF is called the projective limit uniformity on X. It can be shown (Exercise 5) that a sub-basis for f..lF is the set S = (fal (Va) I Va E va for some ex} and that f..lF is the coarsest uniformity on X such that each!a in F is uniformly continuous.
r
Similarly, if g:Y ~ X is a function from a uniform space (Y,v) into X, by Exercise 3 of Section 2.2, g determines a finest uniformity f..l g on X such that g is uniformly continuous. A basis for f..l g is the set B = {U I U is a covering of X such that g-l (U) E v}. Let G = {g a: Y a ~ X Iex E A} be a family of functions from uniform spaces (Ya,v a ) onto X. We define the uniformity f..lG to be in!af..l g ex and call f..lG the inductive limit uniformity on X. It can be shown (Exercise 5) that a basis for f..lG is the set {U IU is a covering of X and g"ixl (U) E Va for each ex} and that f..lG is the finest uniformity on X such that each g a in G is uniformly continuous. PROPOSITION 5.2 The topology generated by the projective limit uniformity is the projective limit topology.
The proof follows immediately from Proposition 5.1 and the definitions of projective limit topology and projective limit uniformity.
EXERCISES 1. Let {'ta } be a collection of topologies for a set X. Show that the set S = u'ta is a sub-basis for SUPa't a . 2. Let F = Ifa:X ~ Y a Ia. E A} be a family of functions from a set X into topological spaces Y a and let 'tF be the projective limit topology with respect to F. Show that the set S = {fal (U a) I U a is open in Y a for some a.} is a sub-basis
114
for 1F and that continuous.
5. Fundamental Constructions 1F
is the coarsest topology on X such that each
fa
in F is
3. Let C = {ga:Ya ~ Xla E A} be a family of functions from topological spaces Ya onto a set X and let 1c be the inductive limit topology with respect to G. Show that B = {U c X I g~.} (U) is open in Ya for each a E A} is a basis for 1C and that 1c is the finest topology on X such that each g a in G is continuous. 4. Let F = If a:X ~ (Y a,v a ) I a E A} be a family of functions from a set X into uniform spaces (Ya,v a ) and let /-!F be the projective limit uniformity with respect to F. Show that the set S = {fal (Va) I Va E Va for some a} is a subbasis for /-!F and that /-!F is the coarsest uniformity on X such that each fa in F is uniformly continuous. 5. Let G = {g a: Y a ~ X I a E A} be a family of functions from uniform spaces (Ya,v a ) onto a set X and let /-!c be the inductive limit uniformity with respect to C. Show that the set B = {U I U is a covering of X and g~l (U) E Va for each a} is a basis for /-!c and that /-!c is the finest uniformity on X such that each g a in G is uniformly continuous.
5.3 Subspaces, Sums, Products and Quotients In this section, the classical constructions will be examined as special cases of inductive and projective limit uniformities. It will be seen that subspaces and product spaces are special cases of projective limit uniformities, while sum and quotient spaces are examples of inductive limit uniformities. If (X, /-!) is a uniform space and Y c X, define i:Y ~ X by iCy) = Y for each y E Y. Let /-!i denote the projective limit uniformity on Y determined by the single function i. It is left as an exercise (Exercise 1) to show that the uniformity /-!i is identical to the subspace uniformity introduced in Section 4.4. Given a collection F = {X a Ia E A} of uniform spaces with uniformities /-!a for each a, we define the uniform product space (P,1t) to be the Cartesian product set P = I1X a (introduced in the Forward) together with the projective limit uniformity 1t determined by the collection {p a Ia E A} of all canonical projections p a:P ~ X a' Recall that the members of P are functions fA ~ uX a such that f( a) E X a for each a. The mappings p a are defined by p a (f) = f( a) for each f E P. When dealing with uniform product spaces such as P = I1X a, it is customary to refer to the individual uniform spaces X a from which the product space is constructed as the coordinate (uniform) spaces. PROPOSITION 5.3 The topology associated with the uniform product space is the topology of the topological product of the coordinate spaces (considered as topological spaces).
5.3 Subspaces, Sums, Products and Quotients
115
Prool: Let'ta be the topology associated with Ila for each a and let 't denote the product topology on P = IlX a' It will suffice to show that for each sub-basic U E 't containing x E P, there exists a V E 1t with S(x,v) c U and for each S(x,W) where W E 1t, there exists a basic V E 't with x EVe S(x,W). If U E 't is a sub-basic open set then U = If E pi/(p) E U ~} for some open U ~ c X~. Since x E U, p~(x) = x(P) E U~ which implies there exists some V~ E /..l~ with S(x(P),V~) c U~. Since 1t is the projective limit uniformity determined by the collection {p a} of canonical projection, by Exercise 5 of Section 5.2, P ~l (V~) is a sub-basic memberof1t. Let/E S(X'Pi31(V~)) = (Pi31(v~)ix(p) E V~ E V~}. Then IE Pi3 1 (V~) for some V~ E V~ that contains x(P). Hence I(P) E V~ C S(x(P),v~) c U ~ so lEU. Therefore, Sex, Pi3 1 (V~)) c U so 't is coarser than the uniform product topology on P. Conversely, suppose W is a basic member of 1t. Since 1t is the projective limit uniformity determined by the collection {p a} of canonical projections, by Exercise 5 of Section 5.2, there exists a finite collection W a1 . .. Wan of coverings in 1la1 ••• /..la n respectively such that W = n7=1Pa~1 (Wa). For each i = 1 ... n choose Va, E W a, with x(a;) EVa,. Then x E V = n7=1 Va, = n7=1P a~l (Va,) and V is a basic member of 'to If I E V then both x and Ibelong to n7=1Pa~1 (Va). But S(x,W) = (IE PiX,fE n7=1Pa~1(W a) for some W a , E W a, for each i = 1 ... n}. Hence x EVe S(x,W) so 't is finer than the uniform product topology on P. Therefore, 't is the uniform product topology on P. Quotient uniformities are an important example of inductive limit uniformities. If (X, /..l) is a uniform space and R is an equivalence relation on X, let q:X ~ X/R be the canonical projection of X onto the quotient set X/R (introduced in Chapter 0). If we then give Q = X/R the inductive limit uniformity /..lq determined by the single function q, then (Q, /..lq) is called the quotient uniform space with respect to R. It is left as an exercise (Exercise 5) to show that the topology associated with /..lq is the topology of the quotient (topological) space X/R, also introduced in the Introduction. A word of caution is in order here. The space Q may not be Hausdorff (see Exercise 7). To insure that the quotient of a Hausdorff uniform space is again Hausdorff, we need the concept of a uniform quotient (see Exercise 8). It is of interest to observe that every onto mapping I:X ~ Y from X to a set Y determines an equivalence relation Rf on X by taking (x,y) E Rf if I(x) = I(y). Moreover, the canonical projection q:X ~ X/Rf determines a one-to-one function g:X/Rf ~ Y such that I = g © q. g is the function defined by g(x*) = I(x) where x* is the equivalence class containing x. If Y is also a uniform space with uniformity v, then by Proposition 0.20, I is continuous if and only if g is continuous. It is easily seen that this continuity condition also holds for uniform continuity; i.e., I is uniformly continuous if and only if g is uniformly continuous. In fact. if g is uniformly continuous, then I must be uniformly continuous since it is the composition of the uniformly
116
5. Fundamental Constructions
continuous function g and the canonical projection q (which is uniformly continuous by definition). Conversely, if f is uniformly continuous then for each V E v, 1(V) E f.l implies that q [{~l (V)] E f.l q (the quotient uniformity on X/R r ). We will show that g~I(V) = qW1(V)] thereby showing g to be uniiormly continuous. For this let V E V and suppose x* E g ~l (V). Then g(x*) =f(x) E V. Let y E 1[((x)]. Then fey) = f(x) which implies y E x* so q(y) = x*. Thus x* E q(f~I(V» so g~I(V) C q(f~l(V». Next suppose z* E q(f~I(V». 1(V) which implies fez) E V. But g(z*) = fez) so z* E g ~l (V). Then Z E Therefore q(f~l (V» c g~l (V) which implies g~l (V) = q(f~l (V».
r
r
r
In Section 5 of Chapter 1, we introduced the concept of the metric space associated with a pseudo-metric space (X, d) by defining the equivalence relation - in X by x - y if d(x,y) = 0 and defining a metric p on X/- by p(x*,y*) = d(p ~l (x*), P ~l (y*» where p:X ~ X/- was the canonical projection and x* and y* were the equivalence classes with respect to - containing x and y respectively. A similar construction can be done for uniform spaces; i.e., if (X,f.l) is a non-Hausdorff uniform space, it is possible to construct a Hausdorff uniform space from X. For this, define R 1.1 on X by (x,y) E R 1.1 if x E S(y,U) for each U E f.l. Then let q:X ~ X/R 1.1 be the canonical projection of X onto the quotient set X/R 1.1 and let f.l q be thelluotient uniformity determined by q. Then (X/R 1.1' f.lq) is a uniform space called the Hausdorff uniform space associated with (X, f.l). PROPOSITION 5.4 The Hausdorff uniform space associated with (X.f.l) determines a Hausdorff topology and the canonical projection q:X ~ XIR 1.1 is both an open and closed mapping. Proof: Let x* and y* be distinct members of X/R I.l' Then y does not belong to x* which implies y does not belong to S(x,U) for some U E f.l. By Exercise 6 of Section 5.2, B = {V I V is a covering of X/R 1.1 and q ~l (V) E f.l} is a basis for f.l q. Now q(U) = {q(U) I U E U} covers X/R I.l and U < q~l [q(U)] so q(U) E f.l q. If y* E S(x*, q( U» then both x* and y* E q(U) for some U E U which implies x,y E U so Y E S(x,U) which is a contradiction. Hence y* does not belong to S(x*,q(U». Let WE f.l q such that W* < q(U). Then S(x*,W)nS(y*,W) = 0 so (X/R >" f.lq) determines a Hausdorff topology.
To show that the canonical projection q is both an open and closed mapping, first let U be an open set in X and suppose x* E q(U). Then q(x) = q(y) for some y E U. Therefore, x E y*. Let V E f.l such that S(y,v) cU. Then x E S(y,V) which implies x* E S(y*, q(V» c q(U). Hence q(U) is open in X/R 1.1 so q is an open mapping. If F is closed in X and if x* does not belong to q(F) then x E F which implies there exists aWE f.l such that S(x,W)nStar(F,W) == 0. Let V E f.l with V* < W. Suppose S(x*, q(V»nq(F)"#- 0. Then there exists some y* E X/R>, such that x* ,y* E q(V) for some V E V and such that y* E q(F). y* E q(F) implies that there exists some z E F with q(z) = q(y) which in tum implies y E z*, so y E S(z, V). Moreover, x* ,y* E q(V) implies there are r, s
5.3 Subspaces, Sums, Products and Quotients
117
E V with q(r) = q(x) and q(s) = q(y) so Y E S(s,V) and x E S(r,V). Consequently, there exist V I, V2, and V 3 E V such that x,r E V I, s,y E V2, and y,Z E V 3 . But then there are WI and W 2 E W with V I UV 2 C WI and V2 UV 3 cW 2 • NOWX,SE WI ands,zE W 2 impliesW l nW 2 "#0,W I cS(X,W) and W 2 C Star (F, W). Hence S(x,W)nStar(F,W) "# which is a contradiction. Therefore, S(x*, q(V»nq(F) = 0. Thus q(F) must be closed so q is also a closed mapping. -
°
Our last example of an inductive limit uniformity is the uniform sum of a collection {(X a, /-1a)} of uniform spaces. We define a new uniform space LX a by first defining the points of LX a to be the ordered pairs (x, a) where x E X a' The mappings i a:X a ---7 LX a defined by i a(X) = (x, a) are called the canonical injections. The uniformity a of LX a is defined to be the inductive limit uniformity determined by the collection {i a} of canonical injections. The uniform space (LX a,a) is called the uniform sum of the uniform spaces X a' The proof of the following proposition is left as an exercise (Exercise 6).
PROPOSITION 5.5 The topology associated with the uniform sum of a collection of uniform spaces is the topology of the disjoint topological sum of these spaces considered as topological spaces.
The canonical injections essentially transfer the uniformities /-la from the spaces X a onto the disjoint "pieces" i a(X a) of LX a and since a is the inductive limit uniformity determined by the i a's, a is the finest uniformity on LX a that makes all the i a 's uniformly continuous.
EXERCISES 1. Show that if (X, /-1) is a uniform space and Y c X, that the projective limit uniformity /-1, on Y determined by the function i:Y ---7 X such that i(y) = Y for each y E Y is identical with the subspace uniformity on Y introduced in Section 4.4. 2. Show that a net in a product space converges to a point p if and only if its projection in each coordinate subspace converges to the projection of p. 3. Show that a net in a product uniform space is Cauchy if and only if the projection of the net in each coordinate subspace is Cauchy. 4. Show that the product complete.
nx a
of complete uniform spaces (X a' /-1a) is
5. Let (X, /-1) be a uniform space and let R be an equivalence relation on X. Show that the topology associated with the quotient uniform space with respect
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5. Fundamental Constructions
to R is the topology of the quotient (topological) space X/R introduced in the Introduction. 6. Prove Proposition 5.5. 7. Show that if (X, /.1) is a non-normal uniform space with the finest uniformity u, that there exists an equivalence relation R on X such that X/R is not Hausdorff. UNIFORM QUOTIENTS A uniform relation in a uniform space (X, /.1) is an equivalence relation R on X such that for each pair of non-equivalent points x,y E X, there exists a sequence {Un} C /.1 satisfying: (1) If xRx' and yRy', then x' and y' are in no common member of U I . (2) If Un +1 E U n +l , there exists a Un E Un such that if p E U n +1 and pRq, then Seq, Un+d C Un.
A uniform quotient is a quotient uniform space X/R where the equivalence relation R is a uniform relation. 8. Show that a uniform quotient of a Hausdorff uniform space is Hausdorff. 9. Show that the canonical projection q:X uniformly continuous.
--7
X/R of a uniform quotient is
UNIFORML Y LOCALLY COMPACT SPACES A uniform space is said to be uniformly locally compact if it has a uniform covering consisting of compact sets. For each countable ordinal a put X a = a + 1 and let '!a be the order topology on X a' Then X a is compact, so X a has a unique uniformity /.1a consisting of all open coverings. Let (L, /.1) be the sum of the collection {(X a, /.1a) I a < WI }. Let R be the equivalence relation on L defined by x - y if and only if x and y belong to the same X a' Let Y = W1 with the order topology. 10. [So Ginsburg and J. Isbell, 1959] Y has a unique uniformity with a basis consisting of all finite open coverings. R is a uniform relation on (L, /.1). L/R = Y. Let q:L --7 Y be the canonical projection. There exists a U E Il such that q(U) = {q(U) I U E U} is not a uniform covering in Y. L is complete, but q(L) = Y is not complete. L is uniformly locally compact.
5.4 Hyperspaces
119
5.4 Hyperspaces Let (X, /.1) be a uniform space and let X' denote the set of all non-void closed subsets of X. If H, K E X' with He Star (K, U) and K c Star (H, U) for some U E /.1, then Hand K are said to be U-close, denoted by IH - K I < U. Note that this relationship is reflexive, i.e., I H - K I < U implies I K - HI < u. If U E /.1 and F E X', put B(F,U) = {K E x'il F - KI < U) and let U' = {B(F,U) IFE X'). Then define /.1* = {U' IU E /.1). We want to show that /.1* is the basis for a uniformity /.1' on X' but first we establish some useful lemmas. LEMMA 5.] If u* < V.IH-FI < U.and IK-FI < U.then IH-KI
<
v.
Proof: IH - F I < U implies H c Star (F, U) and F c Star (H, U). IK - F I < U implies K c Star (F, U) and F c Star (K, U). Therefore H c Star (Star (K. U),U) c StarCK. V). Similarly K c Star (H. V) so IH - K! < V.LEMMA 5.2 If u* < V and FE X' then S(F.U') c B(FY) c S(F.V'). Proof" Let H E S(F.U'). Then H E B(K.U) for some K E X' such that F E B(K,U). Therefore, H c Star(K,U), K c Star(H.U), Fe Star(K,U). and K c Star (F, U), so H c Star (Star (F. U). U) c Star (F. V). Similarly, F c Star (H, V) which implies H E B(F,V). Therefore S(F,U') c B(F,V). B(F,v) c S(F,V') follows from the definition of S(F.V').PROPOSITION 5.6 /.1* is a basis for a uniformity /.1' on X'. Proof: We need to establish (1) and (3) of the definition of a uniform space. We can do this simultaneously by showing that if V'.W' E /.1*, there exists a U' E /.1* with U' <* V'nW'. For this let U E /.1 such that U* < VnW. Let U' c U'. Then U' = B(E,U) for some E E X'. Let F E Star (U',U'). Then F E B(K,U) for some K E X' such that K E B(E,U) which implies IF - K I < U and IK - E I < U. By Lemma 5.1, IF - EI < V so F E B(E,V) E V'. Therefore, Star (U',U') c B(E,v) so U' <* V'. Similarly. U' <* W'so U' <* V'nW'. Thus /.1* is a
basis for a uniformity /.1' on X'. The uniform space (X', /.1') is called the hyperspace of (X, /.1). Hyperspaces of topological spaces have also been studied. The original notion of a hyperspace is due to F. Hausdorff (see Mengenlehre, 3rd edition, Springer, Berlin, 1927). Hausdorff defined a metric on the set of non-empty closed and hounded suhsets of a given metric space. L. Vietoris generalized the concept to topological spaces (see Bereiche Zweiter Ordnung, Monatshefte fur Mathematik und Physik, Volume 33, 1923, pp. 49-62). N. Bourbaki introduced the uniformity for the hyperspace of non-void closed subsets of a uniform space in terms of an entourage uniformity (Topologie General. Paris. Hermann. 1940).
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In this section we translate Bourbaki's approach in tenns of covering unifonnities. PROPOSITION 5.7 The hyperspace of a uniform space is Hausdorff. Proof: Let (X, 11) be a unifonn space and (X', 11') its hyperspace. Choose H, K E X' with H :f. K. Then either H contains a point not in K or K contains a point not in H. Without loss of generality we may assume there is an x E H such that x does not belong to K. Theorem 2.6 can be used to show there exists U,V E 11 with U* < V such that S(x,V)nStar(K,v) = 0. Suppose K E 8(H,U). Then K c Star (H, U) and H c Star (K, U) which implies x E Star (K, U) which is a contradiction. Therefore, K does not belong to 8(H,U) so X' is T I. Since X' is regular, it is also Hausdorff. LEMMA 5.3 A net {Fa} in X' converges (clusters) to some F and only lffor each V E 11. {F a} is eventually (frequently) in 8(F,v).
E
X'. if
Proof: Let V E 11 and choose U E 11 with U* < V. If {F a} converges to F then {F a} is eventually (frequently) in S(F,U'). But by Lemma 5.2, S(F,U') c 8(F,V). Conversely, if for each V E 11, {Fa} is eventually (frequently) in 8(F,V), then it is eventually in S(F,v,) by Lemma 5.2, so {F a} converges to F.LEMMA 5.4 {Fa} is Cauchy in X' if and only iffor each V exists a ~ such that for each a ~~. Fa E 8(F J3,v).
E
11, there
Proof: Let U E 11 with U* < V. Since {F a} is Cauchy, there exists a ~ such that for some HEX', Fa E 8(H,U) for each a ~~. Then for each a ~ ~, Fa c Star (H, U) and H c Star (F a,U). Also, F J3 c Star (H, U) and H c Star (F J3,U), Therefore, Fa C Star (Star (FJ3'U),U) c Star (FJ3,V). Similarly, FJ3 c Star (F a, V) so Fa E 8(F J3,v) for each a ~ ~. The converse it obvious since 8(F J3'V) E V'. -
Let {Fa} be a net in X'. A point p E X, each of whose neighborhoods meets cofinally many of the Fa's is said to be a cluster point of {F a} (in X). PROPOSITION 5.8 The set K of cluster points of a net {F a} in X' is closed. If {F a} converges in X'. it converges to K. Proof: Let p be a limit point of K and let U be a neighborhood of p. Then U contains a point k E K. Since k is a cluster point of {F a}, U frequently meets {F a} which implies p E K. Hence K is closed. Next assume {Fa} converges to F E X'. Suppose x does not belong to F. Then, as pointed out in Proposition 5.7, there is a U E 11 with S(x,U)nStar(F,U) = 0. By Lemma 5.3, IF a} is eventually in B(F,U). Therefore, there is a ~ with Fa E B(F,U) for each a ~ ~ which implies Fa c Star (F, U) for each a ~~. Hence x is not a cluster point of
5.4 Hyperspaces
121
F. Consequently, F c K and there can be no points of K that are not in F, so F =
K.Our first undertaking with hyperspaces will be to characterize supercompleteness (see Section 5.1 for the definition) in terms of the behavior of a certain class of nets in the original space X. In 1988. B. Burdick informed the author that he had shown that cofinal completeness implies supercompleteness and wondered if the reverse implication held. The author pointed out that real Hilbert space (Chapter 1) can be shown to be a supercomplete metric space that is not cofinally complete (see Exercise 1). Thereafter, Burdick discovered the following characterization of supercompleteness that yields the implication: cofinal completeness => supercompleteness as a corollary. Burdick defines a net Ix a}, a E A, in a uniform space (X, IJ.) to be almost Cauchy if for each U E IJ. there exists a collection C of cofinal subsets of A such that for each K E C, Ixy I'Y E K} c U for some U E U, and such that uC is residual in A. Clearly, Cauchy nets are almost Cauchy and almost Cauchy nets are cofinally Cauchy. His characterization of supercompleteness is that a uniform space is supercomplete if and only if each almost Cauchy net clusters. Burdick's proof depends on a theorem of Isbell on partially convergent functions. Isbell's development of partially convergent functions will be given in a later section. For now we present an alternate constructive proof directly from the definitions of supercompleteness and almost Cauchy nets. This constructive proof has the advantage of making clear just how the clustering of almost Cauchy nets is related to supercompleteness. The proof is rather long, so we decompose it into two lemmas plus the main constructions. We define a net IF a} in X' to be proper if IF a} is ordered by set inclusion (a ~ ~ if and only if F 13 c F a) and if Fa C F E X' for some a then F ElF a}. LEMMA 5.5 A proper net IF a} in X' is Cauchy ~ with F 13 c Star(F )jar each a.
V E IJ. there is a
a.v
if and only if for
each
Proof: Assume IF a} is a proper Cauchy net in X'. Let V E IJ.. Pick U E IJ. with U** < V. By Lemma 5.4, there is a 'Y such that for each a : : : 'Y, Fa E B(F y,U). Now CI(Star(F y,U)) = F 13 for some ~ since IF a} is proper. Let a::::: 'Y. Then Fa E B(F y.U) which implies F y c Star (F a, U) so Star (F y,U) c Star(F a,U*), which implies F 13 c Star (F a, V). If a < 'Y then F y c Fa so F 13 c Star (F y' V) c Star (F a, V). Consequently, there is a ~ with F 13 c Star (F a, V)
for each a. Conversely, assume that for each V E IJ. there is a ~ with F 13 c Star (F a, V) for each a. If a : : : 13 then F~ c Star(F a,V) and Fa C FI3 c Star (FI3'V), Therefore Fa E B(F ~.V) for each a::::: 13 so IF a} is Cauchy.-
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LEMMA 5.6 (X, J..l) is supercomplete if and only if each proper Cauchy net in X' converges. Proof: Let {Fa} be a Cauchy net in X'. For each a put H a = u{F 131 ~ ~ a}. Let {S y} denote the collection of elements of X' containing some H a where {S y} is directed by set inclusion (y < 8 if and only if S8 C S y). Clearly {S y} is proper. Since {F a} is Cauchy, so is {S y }. To see this, let V E J..l and pick U E J..l with U* < V. By Lemma 5.4, there exists a ~ such that Fa E B(F 13 ,U) for each a ~~. Therefore, Fa c Star (F 13'U) for each a ~ ~ which implies CI(H 13) c Star (Fj3,V). Now CI(Hj3) = S8 for some Sii E {Sy}. For each y~ 8, Sy C 58 which implies S y c Star (F 13, V). Also, for each y ~ 8, there exists a A ~ ~ with HA c Sy which implies FA c Sy and Fj3 c Star(FA,U) so Fj3 c Star (Sy,U) which implies 5 y E B(F 13'V) for each y ~ 8. Therefore, {S y} is a proper Cauchy net. Then by hypothesis, {S y} converges to its set of cluster points K. Let V E J..l. Pick U E J..l with U*** < V. By Lemma 5.3, {S y} is eventually in B(K,U) so there exists a 8 with S y E B(K,U) for each y ~ 8. Now there exists a ~ with CI(H 13) c 58' Since {Fa} is Cauchy, there exists a ~ with Fa E B(F ~ ,U) for each a ~~. Pick A such that A ~ ~ and A ~~. Then CI(H A) c CI(H 13) c S 8 so CI(H A) = S cr for some (j ~ 8. Hence CI(H A) E B(K,U). Since A ~ ~, Fa E B(F~,U) for each a ~ A which implies Fa C Star(F~,U) and F~ c Star(F a,U) for each a ~ A. Thus CI(H A) C Star(F ~ ,U*) and F ~ c Stare CI(H A)'U*), Hence CI(H A) E B(F ~ ,U*). Consequently, we have
Then multiple applications of Lemma 5.1 yields I Fa - K 1 < V for each a ~ A so {F a} converges to K. Hence X is supercomplete. Burdick's theorem appeared in an article titled A note on completeness of hyperspaces published in the Proceedings of the Fifth Northeast Topology Conference (1991) by Marcel-Dekker. The following proof is due to the author. THEOREM 5.1 (B. Burdick, 1991) A uniform space (X, J..l) is supercomplete if and only if each almost Cauchy net clusters. Proof: Assume each almost Cauchy net clusters and suppose (X, J..l) is not supercomplete. Then by Proposition 5.8 and Lemma 5.6, there exists a proper Cauchy net {F a}, a E A, in X' that does not cluster to its set of cluster points K. There are two cases to consider here. First, K might be the empty set, in which case K would not belong to X' and therefore could not be a cluster point of {F a}. On the other hand, K may not be empty. This is the case we will assume first. Later we will see that an argument similar to the one we will use here can be used to show that under the hypothesis of this theorem, K cannot be the empty set.
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5.4 Hyperspaces
If K *- 0, there exists a U E /-1 such that {Fa} is eventually outside B(K,U) which implies there exists a ~ with Fa not contained in B(K,U) for each a ~~. Let P E K. Then there exists a cofinal C c A with P E Star (F y,U) for each Y E C. Pick 8 E C with 8 > ~. Then F Ii c F ~ which implies P E Star(F~,U). Therefore, K c Star(F~,U) which implies F~ is not contained in Star (K, U) or else F ~ E B(K,U) which is a contradiction. Similarly, Fa is not contained in Star (K, U) for each a ~~. Suppose y is not greater than or equal to~. Then there is a 8 such that y ~ 8 and ~ ~ 8 which implies F 8 is not contained in Star (K, U) and F 8 C F l' Hence F y is not contained in Star (K, U). Consequently, Fa is not contained in Star (K, U) for each a E A. Put U = Star (K, U) and for each a E A let H a = Fa - U. Then IHa} is a Cauchy net in X' that is directed by set inclusion. Let IGy}, yE B, be the collection of elements of X', directed by set inclusion, that contain a memher of IH a}. Then I G y} is a proper Cauchy net in X'. Let D = I (x, y) E X x B 1x E G y} and define ~ on D by (x, y) ~ (y, 8) if Y ~ 8. For each (x, y) E D put \jI(x, y) = x. Then \jI:D ~ X is a net which we will show is almost Cauchy. For this let WE /-1. Pick U E /-1 with U* < W. Since IGy} is stable, there exists a A with G y E B(GA,U) for each y ~ A. Pick x E Star(GA,U). Then some W E W contains Star(x,U). Let V = IW E wlw contains Star(x,U) for some x E Gd. Let R = I(y,~) E 01 ~ ~ A}. Then R is residual in D. Let (y,~) E R which implies ~ ~ A which in tum implies G~ c Star(GA,U) so y E Star(GA,U) which implies y E S(x,U) for some x EGA' Then there exists aWE V containing y. Therefore, \jI(y, ~) = yEW. Let C w = l(y,~)E RI\jI(y,~)E WforsomeWE V}. ThenulCwlWE V}=R. So I C w W E V} is a family of suhsets of D whose union is residual in D such that \jI(C w ) eWE V. It remains to show that C w is cofinal in D for each WE V. For this let WoE V and (y,~) E D. Then there exists an Xo E G A with S(xo,U)cW o. Pick(z,8)E Rwith(xo,A)~(z,8)and(y,~)~(z,8). ThenG8 eGA which implies Z EGA' But (z, 8) E R implies 8 ~ A which in tum implies G 8 E B(G A'U), so G A C Star (G 8,U). Therefore, x E S(s,U) for some S E G 8 which implies S E S(xo,U). Then (xo, A) ~ (s, 8), (y,~) ~ (s, 8), and \jI(s, 8) E S(xo,U) cWo. Hence Cwo is cofinal in D so \jI is almost Cauchy. 1
°
By hypothesis, \jI clusters to some P E X. Let V be a neighborhood of p. Then there exists a cofinal C c D with \jI(C) c V. Let a E A. Then Fa = G ~ for some p E B. Pick x E G~. Then there exists (y, 8) E C with (x, ~) ~ (y, 8) and \jI(y, 8) E V. Therefore, ~ ~ 8 which implies G 8 C G ~ and \jI(y, 8) = y E G 8 C G~ c Fa SO F arN *- 0. Hence p is a cluster point of IF a} so P E K. But \jI c X - U which is closed, so p E X - U which is a contradiction. Next we have the case K = 0. In this case, let D in the argument above be the set {(x, y) E X x A Ix E Fa}. Define:5; on D as before and define \jI:D ~ X by \jI(x, a) = x for each (x, a) E D. Then just as in the argument above, we can show that \jI is almost Cauchy and hence clusters to some p E X. But then as
124
5. Fundamental Constructions
the argument above shows, p E K, so K *' 0. Consequently, (X, /J.) must be supercomplete. Conversely, assume X is supercomplete. Let {xa)' a E A, be an almost Cauchy net in X. For each a put H a = {x~ IB ~ a) and Fa = CI(H a). Then {Fa) isanetinX'. Let VE /J.andpick UE /J.with U*< V. Since {xa) is almost Cauchy, there exists a collection {C y), Y E B, of cofinal subsets of A and a collection {U y) C U such that uyC y is residual in A and {x~ I BE C y} C U y for each y. Pick 0:0 E A such that B E uyC y for each B ~ 0:0. Then H ao C uyC y' For each y pick Vy E V such that Star (Uy,U) C V y' We want to show Fao CUyU y. LetYE Fao' IfYE Hao,clearlYYE UyV y . Supposeyisalimit point of H ao' Let U E U be a neighborhood of y. Then there exists an x ~ E Hao with x~ E U. Now x~ E Hao implies x~ E U y for some y so U C V y. Hence F ao C UyV y. Let a1 E H ao' Then F al C F ao which implies F al C Star (F ao' V). Pick y E B. Let BE C y such that B > a1' Then x~ E U y and x~ E F al which implies x~ E VynF al so uV y C Star(F al ,V). But F ao C UyV y so F ao C Star(F al ,V). Hence F al E 8(F ao,v) for each a1 E H ao so {Fa} is Cauchy. Since X' is complete, {Fa) converges to its set of cluster points K. Let p E K. Let W be a neighborhood of p in X. Then there exists a cofinal C C A with Fan W *' 0 for each a E C. If Y E F anW then y E H a or y is a limit point of H a' In either case, W contains some x~ with B ~ a. For each a E C pick K(a) ~ a such that x K(a) E Fan W. Then {x K(a) Ia E C} c W is cofinal in A. Hence. {x a} clusters top. COROLLARY 5.1 Cofinal completeness implies supercompleteness which in turn implies completeness. COROLLARY 5.2 (1. 1sbell. 1962) If X is paracompact. then it is supercomplete with respect to u. Notice that our proof that supercompleteness implies each almost Cauchy net clusters only relies on the fact that there exists a cluster point of {Fa). Consequently, the existence of a cluster point for each proper Cauchy net in X' is equivalent to supercompleteness. Furthermore, if p is a cluster point for the proper Cauchy net F = {Fa) and U E /J., then for each U E U containing p, UnF ~ *' 0 for cofinally many F ~'s in F. Let y be any index. Then there exists a 8 ~ y with UrlF B *' 0 and since F is directed by inclusion we have UnF y *' 0. Therefore, S(p, U)nF a*,0 for each Fa E F. Thus we can pick an Xu E S(P,U)nF a for each U E /J.. Then {xu} converges to p which implies p is a limit point of Fa for each a. Therefore, p E Fa for each a since each F a is closed. Hence nF a*' 0. Similarly, if nF a*,0 then any p E nF ex is a cluster point of F. We record these observations as
5.4 Hyperspaces
125
PROPOSITION 5.9 If (X. 11) is a uniform space then the following statements are equivalent: (I) (X. 11) is supercomplete, (2) Each proper Cauchy net in X' has a cluster point and (3) For each proper Cauchy net F in X', nF 0.
*'
EXERCISES 1. A function f:X -t Y from a uniform space (X, 11) into a uniform space (Y,v) determines a function j':X' -t Y' where X' and Y' are the hyperspaces of X and Y respectively. j' is defined by j'(A) = Cl y (f1Al) for each A E X'. j' is called the hyperfunction of f. Show that j' is uniformly continuous if and only if f is uniformly continuous. THE HYPERSPACE OF A METRIC SPACE 2. The Hausdorff distance h between two (closed) sets A and B in a metric space (X, d) is defined as the maximum of sup I dCa, B) I a E A) and sup {d(A,b) I b E B) where d(x, S) =in!{ d(x,y) lYE S) for any subset S. Show (a) h is a metric on HX (the hyperspace of X) that generates the topology of HX so that the hyperspace of a metric space is again a metric space. (b) If (X, d) is complete then (HX, h) is complete. THE HYPERSPACE OF A COMPACT SPACE 3. Show that the hyperspace of a compact space is compact. 4. Show that a discrete space of the power of the continuum, with the uniformity determined by all countable coverings, is complete but not supercomplete. 5. Show that real Hilbert space is supercomplete but not cofinally complete. 6. Define the limit inferior of a net {Fa) in X' to be the set inf{F a) = {x E X Ifor each neighborhood U of x, {Fa) eventually meets U}. Similarly, define the limit superior to be the set suplF a) = Ix E X I for each neighborhood U of x, {F a) frequently meets U}. If K is the set of cluster points of {F a} then (a) sup{Fa}=K=inf{F a }
(b) (X, 11) is supercomplete if and only if whenever IF a} is Cauchy then for each U E I..l there is a ~ with Fa C StarCK, U) for each ex ~ ~.
5. Fundamental Constructions
126
7. Use the results of Exercise 8 to show that if X is paracompact, then it is supercomplete with respect to u, without reference to cofinally Cauchy or almost Cauchy nets (i.e., do not appeal to Corollary 5.1).
5.5 Inverse Limits and Spectra We begin our discussion of inverse limits of topological and uniform spaces with a special case that will motivate the concept in general. Let {X n ) be a sequence of topological spaces where n ranges over the non-negative integers and suppose that for each positive integer n, there is a continuous function In:Xn --7 X n- 1 • The sequence of spaces and mappings {Xn,fn) is known as an inverse limit sequence. Inverse limit sequences are often represented by diagrams like the one below:
In
In-!
r::
If m < n, then the composition mapping = fm+1 © fm+2 © ... © fn is a continuous mapping from Xn to X m. Consider the sequence of points lx n ) such that for each n, Xn E Xn and Xn = In+l (x n+!). Then {xn ) can be identified with a point of the product space n;:'=oXn by means of the function g:l --7 uX n, where 1 denotes the non-negative integers, defined by g(n) = Xn- By means of this identification, the set Y of all such sequences can be considered to be a subset of n;:,=ox n. Then Y, equipped with the subspace topology, is called the inverse limit space of the sequence {X n, fn)' Y is denoted by lim.,...Xn or by X~. The functions fn are sometimes called bonding maps.
LEMMA 5.7 If {Xn,fn) is an inverse limit sequence with onto bonding maps and for some counrable set {an) of positive inlegers there is a set {x an } such that xa n E Xa n for each n and such that if m < n, then la~m (x an ) = x am ' then there eXlsrs a point in lim.,...X n whose coordinare in Xa n is xa n for each n. Proof: First assume {an) is infinite. For each positive integer m, there exists a least positive integer an such that an ~ m. If an = m put Xm = x an ' otherwise let Xm = f,:: (x an )· Then IXm) is a point of lim.,...X n such that xa n E Xa n for each fl. Next, assume Ian) is finite. Then there exists a greatest an, say aj. If m < aj put Xm = ;:;. (xa). For each m ~ aj, assume Xm has already been defined. Since fm+l is onto, pick Xm+! E Xm+! such that fm+l (x m+!) = Xm • Then by induction we can complete the sequence IXm) such that Ix m } E lim.,...Xn and for each n, xa n E XOn " Lemma 5.7 illustrates the fundamental property of inverse limit spaces, that for any coordinate Xb all elements of lim"",Xn having that coordinate have
5.5 Inverse Limits and Spectra
127
all other coordinates Xm with m < k determined by the inverse limit sequence, whereas there may be some room for choice of the coordinates Xm with m > k. If the bonding maps are not onto, limf-X n may be the empty set. An example of this occurs when the Xn are all countable discrete spaces, say Xn = {x;:'} and the bonding maps fn:Xn ~ X n -1 are of the form fn(x;:') = x;:,~il' Clearly {Xn> fn} is an inverse limit sequence, but if we begin with a point x~, it is only possible to pick the first m coordinates before we reach xi and there does not exist an X'J'+1 such that fm+l (x'J'+l) = Xl' Consequently, there do not exist any sequences {xnl such that Xn E Xn and fn(x n ) = Xn-l for each n > 1. Therefore, limf-X n = 0. Under certain conditions, we can assure that limf-X n l' 0. For instance. THEOREM 5.2 If each space Xn in the inverse limit sequence /XnJn} is a compact Hausdorff space then limf-X n i= 0. Proof: For each positive integer n let Yn C DXn be defined by {Yi} E Yn if for each j < n, Yj-l =Jj(y). Then limf-Xn = nYn' We will show that for each n, Yn is closed. Suppose p E DXn - Yn- Then for some j < n we have Jj+l (Pi+l) i= Pi' Since Xj is Hausdorff, there exists disjoint open sets Uj and Vj containing Pj and Jj+l (Pj+l) respectively. Put Vj+l = J;ll (Vi) and let Up be a basic open set in DXn containing p and having Uj and Vj+l as its /h and j+ Fi factors respectively. Then no point of Yn lies in Up since if q = {qn} E Up, then qj+l E V j +1 which implies qj E Vj so q does not belong to Up. Therefore, Yn is closed. Since {Yn } is a decreasing chain of closed sets in the compact space DXn, nYn i= 0 so limf-X n i= 0. •
There are many applications of inverse limit spaces of inverse limit sequences in topology. What we are interested in here is extending the concept to uniform spaces, and extending it in a more general setting. For this first notice that by changing the definition of an inverse limit sequence {Xn.fn} so that the Xn are now uniform spaces and the bonding maps are uniformly continuous, we get an inverse limit uniform space limf-Xn that is a uniform subspace of the uniform product space DXno This follows from the parallel between product topological spaces and product uniform spaces [see Section 5.3]. But what we have in mind is something more general than this. Let (D,<) be a directed set and suppose that for each a ED, X a is a topological space. Further suppose that whenever a, ~ E D with a :s; ~, there is a continuous function fl:X i3 ~ X a, and that these functions satisfy the following rules: (1)
(2)
.fl;. is the identity function for each a fl © fr =.ff whenever a < ~ < y.
Let Y= {xala
E
D} andF=
UWI
a, ~
E
E
D
D with a:S; Pl. Then the pair {Y,F}
128
5. Fundamental Constructions
is called an inverse limit system. Since the set of non-negative integers forms a directed set, it is clear that an inverse limit sequence is a special case of an inverse limit system. To define the inverse limit space of an inverse limit system, we let Ix a} denote a net in uX a such that x a E X a for each a E 0 and such that if a < ~ in 0 then .!If (x ~) = X a' Then {x a} can be identified with a point p of nx a having coordinates p a = X a' The collection of all such nets, under this identification, is a subspace of nx a that we denote by lim<-X a and call the inverse limit space of the system {Y, F).
LEMMA 5.8 lim<-X a of the inverse limit system I Y, F) is a closed subspace of the product space nx a. The proofs of Lemma 5.8 and of the following Theorem are left as exercises (Exercises 1 and 2).
THEOREM 5.3 The inverse limit space of an inverse limit system of compact Hausdorff spaces is a compact Hausdorff space, and if each space of the inverse limit system is non-empty, then the inverse limit space is non-empty. Again, if in the definition of the inverse limit system of {Y, F), we require the X a E Y to be uniform spaces and the bonding maps in F to be uniformly continuous functions, we get an inverse limit uniform space lim<-X a that is a uniform subspace of the uniform product space nx a' We now give an important example of an inverse limit uniform space that we will use later on. Let (X,v) be a uniform space and let {
5.5 Inverse Limits and Spectra
129
that U~ < U~, then we put A < 11. Then (A, <) is countably directed. To see this let {An} be a countable set of members of A. For each positive integer k let U~ = U}n ... nU} and put
<1>~
ill
X
~
X)l
.t ·A
II ~
X/
.t <1>~
t)l
<1>~
iA X
-7'
XA
~
X/
where i~ denotes the identity map on X considered as a mapping from X)l onto X A' It is left as an exercise [Exercise 3] to show that <1>~ is uniformly continuous with respect to the metric uniformities on X/
veX) is called the weak completion of X with respect to v. A uniform space is said to be weakly complete if each co-directed Cauchy net clusters. We will show that (X,v) is uniformly homeomorphic to a dense uniform subspace of veX) and that veX) is weakly complete. The weak completion of a uniform space with respect to a uniformity was introduced by K. Morita in 1970 in a paper titled Topological completions and M-spaces published in Sci. Rep. Tokyo Kyoiku Daigaku 10, No. 271, pp. 271-288. To see that veX) is weakly complete, let 'I':D ~ veX) be an co-directed Cauchy net. Then for each A E A,1t A © 'I':D ~ X/
130
5. Fundamental Constructions
belong to CI('VA(R a». Let m be the least positive integer such that S(y,2-m)nStar(CI('V)...(R a», 2- m) = 0. Now for each n > m, y E 'V),JR n) which implies 'V".
contradiction. Therefore,
5.5 Inverse Limits and Spectra
and only
THEOREM 5.4 (K. Morita. 1970) The mappinR <1>.X if X is weakly complete with respect to v.
131 ~
veX) is onto
if
Proof: We first show that if X is weakly complete with respect to v then is onto. For this, let y E veX). Put 'I' = {~1 (1t,Jy» IA E A}. Let {An} be a
countable collection of indicies in A. Since (A, <) is countably directed, there exists ayE A such that An < y for each positive integer n. Therefore, for each n
Consequently, <1>;;1 (1t-y(y» c n;;'=1 ~~ (1tAn (y» so 'I' has the countable intersection property. Also, if U E v, let
5. Fundamental Constructions
132
Therefore, the point y = {YA IA E A} lies in veX). We want to show that the net <1> © ",:D ~ veX) converges to Y so that if <1> is onto then", converges in X (by Proposition 4.8) and hence X is weakly complete with respect to v. Let S(y,1t~l (<1>A(U~))) be a basic open neighborhood of y in veX). Now <1>A(",(R(A,n») c S(YA' <1>A(U~» so 1t~1 (<1>A(",(R(A, n»))) c 1t~1 (S(YA' <1>A(U~)))
C
S(y, 1t~1 (<1>A(U~))).
By the definition of <1>:X ~ veX), it is clear that for each x E X, <1>(x) E 1t~l(<1>A(X» so
-
EXERCISES 1. Prove Lemma 5.8.
2. Let (X,v) be a uniform space and let {Il>A IA E A} be the collection of all normal sequences of open uniform coverings. For each A E A let X A. and X/tf>A. be the pseudo-metric space and the metric space derived from X with respect to
5.6 The Locally Fine Coreflection
133
<1>1.. as constructed in this section. For each pair A, f.l E A with A < f.l, let <1>~ be the bonding map from X/ll into X/'A' Show that <1>~ is uniformly continuous
with respect to the metric uniformities on X/J.l and X/'A' 3. Let (X,v) be a uniform space and let veX) be the weak completion with respect to v. Let <1>:X --j veX) be the function defined in this section by <1> (x ) = {<1>A(x) IA E A I for each x E X. Show that <1> is uniformly continuous. 4. Let {Y, F) be an inverse system of uniform spaces where Y = {X a I a E Dl and F = {fll a, ~ E D with a < ~}. Let C be a cofinal subset of D. Put Z = {X a Ia E C) and G = {fll a, ~ E C with a < ~ I. Show that the inverse limit of {Z,G} is homeomorphic with the inverse limit of {Y, Fl.
5.6 The Locally Fine Coreflection A uniform space is said to have some property P uniformly locally if there is a uniform covering such that each member of this covering has property P. For instance, a uniform space (X, f.l) is said to be uniformly locally compact if there exists a U E f.l such that each U E U is compact. There are a variety of such properties P that have been studied as uniformly local properties. Similarly, a collection {D a} of subsets of X is said to be uniformly discrete if there exists a U E f.l such that for each pair a,~, StareD a,U)nStar(D ~,U) = 0. In a paper titled Some operators on uniform spaces (Transactions of the American Mathematical Society, Volume 93, pp. 145-168), published in 1959, S. Ginsburg and J. Isbell introduced the notion of locally fine uniform spaces, and the concept of the locally fine corefiection (a finer uniformity) associated with a given uniformity. To define these concepts, they made use of the notion of a uniformly locally uniform covering of a uniform space (X, f.l). A covering V of X is said to be uniformly locally uniform if there exists a U E f.l such that for each U E U, the covering {Un V IV E V} of U is a uniform covering (in the induced subspace uniformity on U). This means that if U = {Va} then for each fixed a, we may replace V with a uniform covering Va = {vg} such that {VanVgl = Va I Va, the trace of Va on Va (Va restricted to Va). A uniform space (X, f.l) is then defined to be locally fine if each uniformly locally uniform covering belongs to f.l. It is easily seen that (X, f.l) is locally fine if and only if it is closed under what Ginsburg and Isbell call the staggered intersection operation {V anVg} where {Val E f.l and {vg} is a uniform covering of Va for each a. Next, they constructed a new uniformity A(f.l) for X, that is the coarsest uniformity for X finer than f.l, that is locally fine. The uniform space (X, A(f.l» is called the locally fine coreflection of X (the meaning of the term corefiection will be explained in the next section - it is not necessary for understanding the uniformity A(f.l».
134
5. Fundamental Constructions
To define A(j..l) we proceed via transfinite induction using the concept of the derivative of a uniformity. Let j..l0 = J.! and define the derivative j..li of j..l to be the family of all coverings which have refinements of the form (U o.nVg:), where (U a I E j..l and for each fixed a, I Vg:) E j..ll U u' It is easily shown that j..li is a filter of coverings with respect to refinement, and that j..l0 C j..ll. Then, for each ordinal number a, we can define j..lo. by the inductive rules 110.+1 = (j..lU)! for all ordinals a and for limit ordinals B by j..l~ = u (j..lu Ia < B). Clearly, there must exist a last derivative 11K = j..lK+1. We put A(j..l) = 11K. It is easily shown that A(j..l) is a filter of coverings with respect to refinement. It is considerably more difficult to show that A(f.l) is a uniformity for X. The proof given by Ginsburg and Isbell (1959) and the revised proof given by Isbell in his book (1964) are not entirely correct. The proof given below incorporates the idea of the lemma they tried to prove prior to the theorem in order to avoid some of the problems with their proofs.
THEOREM 5.6 For a complete metric space (X, j..l) where j..l is the metric uniformity, A(j..l) = u.
Proof: By Stone's Theorem (Chapter I), u consists of all coverings that have open refinements. Each member of j..l has an open refinement. This property is preserved under the operation of taking the derivative. Consequently, A(j..l) C u. To show the converse, let V = (V ~) be an open covering of X. For each positive integer n, let Un = {U~} be a uniform covering of X consisting of the spheres of radius 2- n • Let P be the set of all indices (n, a) of the sets U~ in the covering Un and < be the partial ordering on P defined by: (n+ 1, B) < (n, a) if Ur- l nU~ *- 0, and (n+k, y) < (n,a) if there exists a chain (n+k, y) < (n+k-I,8) < ... < (n, a). Let S be the set of all (n. a) with Star(U~,Un) C V ~ for some B. Since (X, j..l) is complete and each U~ is of radius 2- n , each infinite descending chain in P converges to some limit point x. Since some £-sphere of x is contained in some V 13, only finitely many of the U~ in the chain can have indices in the chain P - S. We now define an ordinal valued function 8 on P - S as follows: if every predecessor of p is in S then 8(p) = O. If 8(q) is defined for each predecessor q of p in P - S, then 8(p) = a where a is the least ordinal greater than all these 8(q). First. we observe that this defines 8 on all of P - S, for otherwise there would exist some PiE P - S on which 8 is not defined which implies there is some predecessor P 2 < P 1 in P - S for which 8 is not defined, and so on, so we get an infinite descending chain ... < Pn < ... < P 2 < P 1 of elements of P - S for which 8 is not defined which is a contradiction. Next, we show that for each (n,a) E P - S, {U~nU~ I(k,Y) E S} E A(j..l) I U~. For this we induct on the values of 8. To get the induction started (and to prove another result we will need later), let p either belong to S or have o(P) == O. In either case, each predecessor of P is in S. Let Q 0 = {s E sis is an immediate
5.6 The Locally Fine Core flection
135
predecessor of pI, and for each non-negative integer n let Qn +1 = {S E sis is an immediate predecessor of some q E QIl I. Then uQn is the set of all predecessors of pin S. Now let U~ = {UpnUsls E Qol E ~Iup = ~o on Up. Then Uf = {Up nUs Is E U'~l Q;} E ~l on Up and ug c Uf. Continuing this process by induction, we obtain a sequence of coverings {U~ I = {Up nUs Is E Ui~nQi I with U~ c U~+l and U~ E ~n for each non-negative integer n. Consequently. U~ = {UpnUs Is E uQn I E A(~) IUp. Now P = (n,a) for some non-negative integer n and some index a. Then for each positive integer m such that m < n, if (m, '() does not precede (n. a) then U~nU;' = 0. Hence if s E S with s not in uQn and UDnU S ::F 0 then s = (k,~) for some non-negative integer k with k ~ n and some index~. But then U~ E Uk. Since there are only finitely many Uk with k ~ n we see that {UpnU, Is E SI E A(~) i Up. To keep the induction going, we assume {UpnU,ls E SI E A(~)iUp for each pEP with 0(P) < K for some ordinal K. Let q E P - S with o(q) = K. We maintain that {UqnUs Is E SI E A(~) IUq. To see this, note that {UqnUs Is E SI is refined by {(UqnUr)nUs Ir is an immediate predecessor of q and s E S}, that {UqnU r Ir is an immediate predecessor of q I E ~ IUq and for each r < q, {UrnU, I s E S I E A(~) IUr by our induction assumption. To see this last assertion note that if r < q and rEP - S then oCr) < o(s), whereas if rES, then all predecessors of r are in S, which is one of the cases we have already proved. Consequently, {UqnUsls E SI E A(~)IUq, so {UpnUsls E S} E A(~)IUp for eachp E P - S. Now {Up Ip is a maximal element in PIE ~ since if p is maximal, then p
=
(l,a) for some index a. For each maximal p, either PES or pEP - S. If pEP
-Sthen {UpnUslsE SI E A(~)lup. IfpE S,theneachpredecessorofpisinS so {UpnUsls E SI E A(~)lup. Therefore, U = {Usls E SI E A(~) by definition of A(~). But clearly, U < V, so V E A(~). Hence u c A(~), so A(~) =
u.PROPOSITION 5.10 For each U E 11. where (X,~) is a uniform space, there exists a normal sequence {Un) oj locally filllte open members ojA(~) with U j < U. Proof' If U E ~. there exists a uniformly continuous j:X ~ M where M is a metric space, and a uniform covering V of M with 1 (V) < U (Theorem 2.5). Let M' be the completion of M. Then;:X ~ M' is uniformly continuous and l (V') < U where V' is the Morita extension of V to M' (see Exercise 4, Section 4.4). Since M' is paracompact, there exists a normal sequence {V~ I of locally finite open coverings of M' with V'l < V'. By Theorem 5.6, each V~ E A(V) where v is the uniformity of M'. Suppose V~ E va for some ordinal a. By a straightforward induction argument, we can show that Un = 1 (V~) E ~a. Hence Un E A(~) for each n. Clearly Un is locally finite for each nand {Un I is a normal sequence of open, locally finite members of A(~) with U 1 < U.-
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PROPOSITION 5.11 For a uniform space (X, f..l), A(f..l) is a uniformity for X. E A(f..l), there exists a normal sequence {Un I of open, locally finite members of A(f..l) with U 1 < U. By Proposition 5.10, it is true for members of j..I.. Assume a is the least ordinal for which the assertion fails for some U E j..I.u. Now U = {V~n W~ I where {V p} E j..I. K for some K < a and for each index P, {W~ I E f..l K( ) for some K(P) < a. Hence {V p I has an open locally finite refinement {H ol. E A(f..l) with an open locally finite star refinement (K e) E AC/..t). For each 0 choose a P(o) with Hoe V P(o). Since {H 0 I is locally finite, each K e is contained in only finitely many of the H 0, say H 01 . . . H On' Put We = {w~(81l In ... n {w~(8n) I. Since each f..lu is a filter, it can be seen from the induction hypothesis that We has an open, locally finite, star refinement I that is normal with respect to the collection of open, locally finite members of A(f..l). But then, (K I is an open, locally finite, star refinement of U that can be shown to be normal with respect to the open, locally finite members of A(f..l). Therefore, the assertion does not fail after all, so A(f..l) is a uniformity. -
Proof: We will prove inductively that for each U
{Z!
enZ!
COROLLARY 5.3 The uniformity A{f..l) has a uniformly locally finite basis. THEOREM 5.7 For a uniform space (X, f..l). A(f..l) is the coarsest locally fine uniformity finer than j..I.. Moreover, each locally fine uniformity has a basis of uniformly locally finite coverings. Proof: By construction, A(f..l) is locally fine, and finer than f..l. If v is another locally fine uniformity finer than f..l, then it is finer than f..ll. By a straightforward induction argument, v is finer than j..I.u for each ordinal a and hence finer than A(f..l). Therefore, A(f..l) is the coarsest locally fine uniformity finer than f..l. Finally, if f..l is locally fine, then A(f..l) = f..l, so each U E f..l has an open uniformly
locally finite refinement. Therefore, each locally fine uniformity has a uniformly locally finite basis. Let P be a partially ordered set. ReP is residual in P if for each r E R, s R for each s E P with r < s. Note that the restriction R ::f- 0 has been dropped from the corresponding definition for directed sets. A function ",:P ~ X is called a partial net (denoted p-net) in X. A p-net '" in a uniform space (X,v) is partially Cauchy if for each {U u} E v, there exists a family {R u} of residual subsets of P such that for each a, ",(R u) c U U and uR u is cofinal in P. The definition of a cofinal subset of P is the same as the definition for directed sets. We extend this definition by allowing v to be f..lu for some uniformity f..l and ordinal a. In this case, we say", is partially Cauchy with respect to the a 1h derivative of f..l. '" is said to partially converge to some x E X if for each neighborhood U of x, "'-' (U) contains a non-empty residual subset of P. If Pis E
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a directed set, the definitions of partially Cauchy and partially convergent are the same as Cauchy and convergent respectively. LEMMA 5.9 If'l':P ~ (X, Il) is partially Cauchy, so is 'I'.P ~ (X,A(Il)). Proof: Suppose 'I':P ~ (X, A(Il)) is not partially Cauchy. Then there exists a least ordinal ex for which 'I':P ~ (X, IlU) is not partially Cauchy. ex must be a non-limit ordinal, say ~ + 1, for otherwise Ilu is just the union of its
predecessors, for which 'V is partially Cauchy. Let {U ynYS} E f,.lu. Then {U y } E Il~ and for each y, {YS} E Il~. For PEP, there exists a successor q, all of whose successors are in one of the U y' For this y, q has a successor r, all of whose successors are in one of the YS. Hence there exists a residual R(y,8) 1= 0 with 'I'(R(y,8)) c U ynYS such that p .,::; r for each r E R(y,8). But then there exists a cofinal union of residual sets, each mapped by 'V into one of the U yn YS. For each U ynYS not containing one of the.se residual images of 'V, put R(y,8) = 0. Then 'V is partially Cauchy with respect to Ilu which is a contradiction. Therefore, 'I':P ~ (X, A(Il)) is partially Cauchy.· The proof of the following theorem and the terminology used in the proof is somewhat different from Isbell's original proof, but his central ideas are preserved. This different route frees us from having to introduce additional terminology and the concept of a stable Cauchy filter, and from proving some additional lemmas. THEOREM 5.8 (1. Isbell, 1962) For a uniform space (X, Il), the following are equivalent: (1) (X, Il) is supercomplete. (2) X is para compact and A(Il) is fine. (3) Each partially Cauchy p-net in (X, Il) partially converges. Proof: (1) ~ (2) Condition (2) implies each open covering belongs to A(Il). Suppose this is not the case, i.e., suppose the open covering U is not A(Il)-uniform. Let (X', III be the hyperspace of (X, f,.l). We construct a proper
Cauchy net in X' that has no cluster point. By Proposition 5.10 this is a contradiction. For this let B consist of those closed sets F c X with the trace of U on some uniform neighborhood of X - F being A(Il)-uniform. Let F, K E B. Then FnK is closed. F, K E B implies there exists V, WEll with U being A(Il)-uniform on Star(X - F,V) and U being A(Il)-uniform on Star(X - K,W). Let Z = VnW. Then Z Eiland U is A(f,.l)-uniform on both Star(X - F, Z) and Star(X - K, Z). If FnK = 0 then F c X - K so U is A(Il)-uniform on Star(F, Z). Therefore, for each Z E Z, U IZ E A(Il) Iz so U E A(f,.l) which contradicts our assumption, so FnK 1= 0. Also, since U is A(f,.l)-uniform on both Star(X-F,z) and Star(X-K,z), U is A(Il)-uniform on Star(X-F,z)uStar(X-K,z) which contains Star(X-FnK,z). Consequently, FnK E B.
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Let D be the collection of closed sets in X that contain a member of Band define:-O; on D by F:-O; K if KeF. Then (D, :-0;) is a directed set. For each FED put \jI(F) = F. Then \jI:D -? X' is a proper net in X'. We need to show that \jI is also Cauchy. By Lemma 5.5, we need to show that for each V E fJ. there exists an FED with \jI(F) c Star(\jI(K},v} for each KED. For this let WE fJ. with W* < V and put F = Cl( {W E wi U is not A(fJ.)-uniform on W}). Then FEB since the union of the W E W on which U is A(fJ.)-uniform is a uniform neighborhood of X - F containing A = Star(X - F,W). To see this let W E W with Wn(X - F) = 0. Then W is not a member of W on which U is A(fJ.)-uniform. Hence W c A. For each KED, there exists H E B with H c K. Star(H,W) contains a dense subset of F, for if x E U {W E wi U is not A(fJ.)-uniform on W} then x E WOE W on which U is not A(fJ.)-uniform which implies W 0 is not contained in X - H (since H E B) so Woe Star(H,W) which implies x E Star(H,W). But then F c Star(H,W*) c Star(K,V). Since \jI(F) = F and \jI(K) = K we have \jI(F) c Star(\jI(K},v} for each KED so \jI is Cauchy. So \jI:D -? X' is a proper Cauchy net. By Proposition 5.10, \jI has a cluster point P E X. Then there exists a U E U with P E U. Pick V E fJ. with S(p,V*) c U and put F = X - S(P,V). Then F is closed and Star(X - F,V) c S(p,V*) c U so U is A(fJ.)-uniform on Star(X - F,v) since U IS(p,v*) = X IS(p,v*). Hence F E D and S(p,v)nF = 0. Then for each KED with K?: F, KnS(p,v) = 0 so pis not a cluster point of \jI which is a contradiction. Therefore, (1) -? (2). (2) -? (3). By Lemma 5.9 we need only show that a partially Cauchy partial net in (X, A(fJ.» partially converges. Assume this is not the case, i.e., that there exists a partially Cauchy partial net \jI:P -? (X, A(fJ.» that does not partially converge. Then there exists an open covering U of X such that \jI-l (U) contains no non-empty residual subset of P for each U E U. By (2), A(fJ.) = U and since X is paracompact, U E u. Hence \jI is not partially Cauchy which is a contradiction. Therefore, (2) -? (3). (3) -? (1) Let \jI:D -? X' be a proper Cauchy net in the hyperspace (X', fJ.') and suppose \jI has no cluster point. Let P be the collection of subsets S of X that are uniform neighborhoods of sets that meet each \jI(d) for d E D. Then P is partially ordered by set inclusion. For each PEP pickf(P) E P. Thenf:P -? X is a partial net. Let x E X. Since x is not a cluster point of \jI, there exists a U E fJ. and 8 E D with Sex, U*)n\jl(8) = 0. Then Q = Star(\jI(8) , U) E P and S(x,U)nQ = 0. Suppose fpartially converges to x. Then there exists a residual ReP with feR) c S(x,U). Let R E R. Now R = Star(T,v) for some V E fJ. and some T that meets \jI(d) for each d E D. Put A = \jI(8)nT and let W = UnV E fJ.. Then B = Star(A,W) E P and B c QnR. Therefore, R :-0; B and feB) E Q which implies feB) does not belong to sex, U) which is a contradiction. Hence f cannot partially converge to any x E X.
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Let {U a} E Jl. If S E P, there exists a V E Jl with Star(T,v*) c S and V** < {U a}. By Lemma 5.5, there exists a 8 E D with ",(8) c Star(",(d),V) for each dE D. Let x E ",(8)nT. Some U y contains R = S(x,V*) c Star(T,V*) c S. Since x E ",(8), each ",(d) meets S(x,V). Hence REP. ReS implies S ~ R in P and all successors of R in P are mapped into U y by f. Therefore, f:P ~ X is partially Cauchy and hence must partially converge by (3). But this is a contradiction, so each proper Cauchy net in (X', Jl) clusters. By Proposition 5.10, (X, Jl) is supercomplete.· In the sense expressed in Theorem 5.8, the uniformity '-(Jl) characterizes supercompleteness in paracompact uniform spaces (X, Jl). In a yet unpublished paper, B. Burdick shows that there exists another uniformity S(Jl) that characterizes cofinal completeness in the same way. We now develop a useful alternate construction of '-(Jl). We denote this new construction by f(Jl) for the time being. I(Jl) is called the subfine corejiection of Jl. To define I(Jl), we first need to define what a subfine uniform space is. A uniform space is said to be subfine if it is a subspace of a fine uniform space. Subfine spaces are a generalization of fine spaces whose definition is much more intuitive than the definition of the locally fine spaces. At the time Isbell published his book in 1964, it was an open problem whether locally fine spaces are subfine. In a 1987 paper titled Locally fine uniformities and normal covers (Czechoslovak Math. Jour. 37(112), pp. 181-187), J. Pelant solved the problem with an affirmative answer. Pelant's complex proof, together with other proofs he references would take up more space than we can allot to this subject. The interested reader can find his proof and the necessary references in the citation above. In what follows, it will first be shown that every uniform space (X, Jl) has a coarsest subfine uniformity I(Jl) finer than Jl, and that by Pelant's Theorem, I(Jl) = '-(Jl). For the construction of I(Jl), we need the concept of an injective uniform space. A uniform space Y is injective if whenever A is a uniform subspace of X, every uniformly continuous function f:A ~ Y can be extended to a uniformly continuous function F:X ~ Y. THEOREM 5.9 The closed interval [0,1] is an injective space. Proof: Let i:A ~ X be a uniform imbedding and f:A ~ I = [0,1] a uniformly continuous function. Let Jl be the uniformity of X and Jlp the collection of all coverings of X refined by finite members of Jl. It is left as an exercise to show that Jlp is a uniformity for X that is coarser than Jl (Exercise 6). Then the identity function jx:(X, Jl) ~ (X, Jlp) is uniformly continuous. Let n:X ~ JlpX be the uniform imbedding of (X, Jlp) into its completion JlI3X,
Now Jl131 A is a uniformity on A coarser than Jl so the identity function ~ (A, Jlp) is .uniformly continuous and Cl l1Bx (A) is the completion of (A, Jlp) since completIons are unique. So Jli3A is a closed subspace of Jli3X, Since Jli3 is a precompact uniformity, by Proposition 4.19, Jli3X is compact.
h :(A, Jl)
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Since I is precompact,f:A --+ I is also uniform continuous with respect to I-l~ IA. By Theorem 4.9, f can be extended to a uniformly continuous function f~:I-l~A --+ I. Then since I-l~A is closed in I-l~X and I-l~X is normal by Proposition 0.32, we can apply Tietze's Extension Theorem (0.2) to get a continuous extension g:I-l~X --+ I. Since both I and ~~X are compact, their unique uniformities are fine, so g is uniformly continuous. Define F: X --+ I by F = g © n: © h. Then for each a E A, F(a) = g(n:(jA (a») = g(a) =f~(a) =f(a), so F is an extension of J. F is uniformly continuous since it is the composition of uniformly continuous functions. Hence [0,1] is an injective space.COROLLARY 5.4 Any closed interval! c R is an injective space. PROPOSITION 5.12 Each product of injective spaces is injective. Proof: If Y = nY a is the product of injective uniform spaces Y a and the uniform subspace A c X is mapped uniformly continuously into Y, then the functions fa = p a © j:A --+ Y a (where p a is the a th coordinate projection) is a uniformly continuous function, so it can be extended to a uniformly continuous function g a: X --+ Y a . Then the function g: X --+ Y defined by g(x) = {g a (x) } extends f:A --+ Y, since f(a) = {fa(a)} for each a E A. To see that g is uniformly continuous, let U be a uniform covering of ny a' Then there exists a finite collection al ... an of indices and uniform coverings Val ... Van of Y a ] ... Y an respectively such that Pa~ (Va] )n ... n Pa~ (Va.) < U. Then
which is uniformly continuous in X since ga~ (Va) is uniformly continuous for each i = 1 ... n. Therefore, g -I (U) is a uniform covering of X so g is uniformly continuous. Consequently, Y is an injective uniform space. A uniform covering U of a uniform space X is said to be realized by a uniformly continuous function f:X --+ Y if for some uniform covering V of Y we have 1 (V) < U.
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PROPOSITION 5./3 If a family of uniformly continuous functions f a: X --+ Y a realizes every uniform covering of X, then the mapping f:X --+ nY a defined by f(x) = (f a (x)) for each x E X, is an imbedding. Proof: Clearly fis uniformly continuous by an argument similar to the one used to prove that g is uniformly continuous in Proposition 5.12 above. To show that fis an imbedding, we need to show that for each uniform covering U ofX,/(U) is a uniform covering of ny a . To see this, note that there exists an index y for U and a uniform covering V of Y y such that (V) < U implies (p y © f)-I (V) <
f:/
5.6 The Locally Fine Coreflection
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U which in tum implies J (Prj (V» < U. But Prj (V) is a sub-basic uniform covering of OY a and since f is uniformly continuous J (Prj (V» is a uniform covering of X. Hence feU) = PrJ (V) which is a uniform covering of OY a' -
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THEOREM 5.10 Every uniform space can be uniformly imbedded in a product of metric spaces. Proof: By Theorem 2.5, for each uniform covering U a of a uniform space X, there exists a uniformly continuous function fa that maps X onto a metric space M a such that the inverse image of each set of diameter < 1 is a subset of an element of Ua. Therefore, U a is realized by fa. Then by Proposition 5.13, the mapping f:X ~ OM a defined by f(x) = {fa(x)} for each x E X is an imbedding.
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For a set S, the metric space I ~(S) of all bounded real valued functions on S with the distance function d(f, g) = sup{ If(x) - g(x) II x EX}, is called the I ~ -space of S. THEOREM 5.11 Each metric space can be uniformly imbedded in the unit sphere of an 1~-space(S)for some set S.
Proof: Let (M, d) be a metric space. For each pair x,y E M put 8(x,y) = min {d(x,y),l }. Clearly (M, 8) is uniformly homeomorphic with (M, d). For each x E M, let x* be the function on M defined by x*(y) = 8(x,y). Then d(x*,y*) = sup{ 18(x,z) - 8(y,z)llz EM}. For any z E M, the largest 8(x,z) can be is 8(x, y) + 8(y, z) because of the triangle inequality. Therefore, sup { 18(x, z) - 8(y, z) II Z E M} ~ sup { 18(x,y) + 8(y. z) - 8(y, z) II z E M} = 8(x,y) so for each pair x,y E M, d(x* ,y*) ~ 8(x,y). Conversely, Ix*(Y) - y*(y) I = 18(x,y) - 0 I = 8(x,y) so sup{ Ix*(z) - y*(z)llz E M} = 8(x,y). Hence, for each pair x,y E M, d(x* ,y*) = 8(x,y), so (M, 8) is uniformly homeomorphic with f(M, 8) c I ~(M) where f:M ~ I ~(M) is the uniform imbedding defined by f(x) =x* for each x E M. Since (M, d) is uniformly homeomorphic with (M, 8), we have (M, d) is uniformly homeomorphic with f(M) c I ~ (M). Since Ix*(m) I ~ 1 for each m E M, (M, d) can be uniformly imbedded in the unit sphere of an I ~-space. PROPOSITION 5.14 The unit sphere of an I ~-space is injective. Proof: Let S J (X) denote the unit sphere of the I ~ -space I ~ (X). Let Z be a uniform space andf:Z ~ S J (X) a uniformly continuous function. We first show that f is uniformly continuous if and only if the function g:Z x X ~ [-1,1] is uniformly continuous where g is defined by g(z, x) = [f(z)] (x) and X is considered to be uniformly discrete. For this, suppose f is uniformly continuous and V is a uniform covering of [-I,ll. Then there exists a uniform covering U of Z such that f maps any two points that are U-close in Z into two functions that are V-close in S J (X). Then the product U x W where W = { {x} Ix EX}
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is a uniform covering of Z x X. Moreover, U x W is finer than g(V*). To see this note that g(V*) = {g(Star(V,V» I V E VI. For each V E V, g(Star(V,v» = {(z,x) E Z x xl [f(z)](x) E Star(V,V)}. Now if (ZI ,x) and (z2. x) E U x {xl for some U x {x} E U X W. then z 1 and z 2 are U-close in Z which implies t(z ") andf(z2) are V-close in S 1 (X). so [fez J )l(x) and [f(Z2)](X) are V-close in [-1J I. Hence [f(zd](x), [f(Z2)](X) E Vo for some Vo E V, so g(l! x {x)) c Star(V" , V). Therefore, U x W is finer than g(V*) so g is uniformly continuous. Conversely, suppose g is uniformly continuous. Then there exists a uniform covering U of Z and a uniform covering W of X such that U x W < g(V). Without loss of generality we may assume W = { {x} Ix EX}. Pick x IJ E X. If zl, z2 E U for some U E U. then (z 1. xo). (Z2. xo) E U x {xo} c g(V) for some V E V. But g(V) = {(z, x) E Z X X I [f(z)](x) E V} so [fez 1 )](x), [f(Z2)](X) E V. Since this holds for each Xo in X, f(zd and f(z2) are two functions in S 1 (X) that are V-close. Hencefis uniformly continuous. To show that S 1 (X) is an injective uniform space. let A be a uniform subspace of Y and let f:A ~ S 1 (X) be a uniformly continuous function. Then by the above result, g:A x X ~ [-1,1] is uniformly continuous where g(a, x) = [f(a)](x). g can be extended to a uniformly continuous functIOn G:Y x X ~ [-1,1] (this follows from Theorem 4.9 and Exercise 4 of Section 2.2). But then F:Y ~ S 1 (X) is uniformly continuous where for each Y E Y, F(y) is the function in S 1 (X) defined by [F(y)](x) = G(y. x). Clearly, F is an extension of j, so S 1 (X) is an injective space. • THEOREM 5.12 Each uniform space can be uniformly imbedded in an injective uniform space. Proof: Let X be a uniform space. By Theorem 5.10. X can he imhedded in a product I1M u of metric spaces. By Theorem 5.11, each M u can be imbedded in the unit sphere S 1 (M u) of the I ~-space I ~(M u). Consequently, X can be imbedded in OS 1 (M u). By Proposition 5.14, S 1 (M u) is an injective uniform space for each a. Then by Proposition 5.12, OS 1 (M u) is an injective uniform space, so X can be imbedded in an injective uniform space. •
For a uniform space (X, ~), let (Y. v) be an injective uniform space in which (X, ~) can be imbedded. Let u be the finest uniformity on Y and put l(~) = u I X. Clearly (X, l(~» is a suhfine uniform space. The next lemma shows that l(~) is well defined. l(~) is called the subfine coreflection of ~. LEMMA 5.10 If (X, ~) is uniformly imbedded In an injective uniform space (Z, ~), and if u is the fine Uniformity on Z, then (X, l(~)) and (X, m(/l)) are uniformly homeomorphic where m(/l) = u Ix. Proof: Let f:A ~ (X, ~) be uniformly continuous where A is a subspace of a fine uniform space B. Thenfhas an extension g:B ~ (Y, v) since Y is injective
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and (X, /-1) is imbedded in (Y, v). Since B is a fine space, g is uniformly continuous into (Y. u) where u is the fine uniformity on Y. But then the restriction of g to A is uniformly continuous, so f:A ~ (X, 1(/-1» is uniformly continuous. Therefore, any uniformly continuous function f from a subfine space into (X, /-1) is also uniformly continuous into (X, [(/-1». The identity function i:(X, m(/-1» ~ (X, /-1) is uniformly continuous, so by the above result, i:(X, m(/-1» ~ (X, [(/-1» is uniformly continuous. If (Y, v) were replaced with (Z, 0 in the above argument, we would obtain the result that any uniformly continuous function f from a subfine space into (X, J.1) is .also uniformly continuous into (X, m(/-1». Hence i-I :(X, 1(/-1» ~ (X, m(J.1» is uniformly continuous, so (X, m(/-1» is uniformly homeomorphic with (X. 1(/-1». -
COROLLARY 5.5 The sUbfine corejiection of an injective space or a complete metric space isfine. Proof: The injective case is clear from the proof of Lemma 5.10. If (X, /-1) is a complete metric space, then by Theorem 5.11 and Proposition 5.14, (X, /-1) is a closed subspace of an injective metric space (Y, v). Since X is closed in Y, each open covering of X together with the set Y - X is an open covering of Y. Hence the fine uniformity on Y induces the fine uniformity on X.LEMMA 5.11 If(X. /-1) is sUbfine then it is locally fine. Proof: Assume (X, /-1) is subfine. It suffices to show that 1-(/-1) = /-1. For this let (X, /-1) be uniformly imbedded in some fine space (Y, v). Then /-1 = v I X. Then as shown in the proof of Lemma 5.l0, the identity function i:(X, /-1) ~ (X. [(/-1» is uniformly continuous. Clearly i-I :(X, [(J.1» ~ (X, 11) is uniformly continuous since [(/-1) is finer than J.1. Therefore, (X. /-1) is uniformly homeomorphic with (X,[(/-1» so J.1 = [(/-1).-
LEMMA 5.12 [(/-1) is the coarsest sUbfine uniformity for Xfiner than J.1. In particular. /-1 is sUbfine if and only if [(/-1) = /1. Proof: Suppose there exists a subfine uniformity 8 for X such that /-1 C 8 C [(/-1). Then i:(X, 8) ~ (X, /-1) is uniformly continuous, so by the proof of Lemma 5.l0, i:(X, 8) ~ (X, 1(/-1» is uniformly continuous. Clearly i-I :(X. [(/-1» ~ (X, 8) is uniformly continuous since 1(/-1) is finer than 8. Therefore, (X, [(11» is uniformly homeomorphic with (X, 8), so 1(/-1) is the coarsest subfine uniformity for X finer than /1.LEMMA 5.13 A point finite uniform covering has a uniformly locally finite uniform shrinking. A a-disjoint uniform covering has a a-discrete uniform shrinking.
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Proof: Let {Va} be a point finite unifonn covering. By Exercise 6 of Section 2.2, there exists a unifonn covering {Va} that is a unifonnly strict shrinking of {Va}. Let W be a uniform covering such that for each a, Star(V a , W) eVa. Let p E X and pick W E W with pEW. If Wn V 13 1:- 0 then p E V 13' Consequently, W can meet at most finitely many of the Va's since p can be in at most finitely many of the Va's. Therefore, {Va} is unifonnly locally finite.
If {Va} is a a-disjoint unifonn covering, there exists countably many subcollections {VB} c {Va} such that each {VB I is a disjoint collection. Since for each a, Star(V a , W) eVa, the collection {VB} is unifonnly discrete. Hence {Va} is a a-unifonnly discrete unifonn shrinking of {Va}. LEMMA 5.14 Each uniform covering of a sUbfme uniform space has a locally finite uniform refinement and a a-disjoint uniform refinement. Proof: Let (X, 11) be a subfine uniform space. Then there exists a fine unifonn space (Y, 11*) such that (X, 11) is a subspace of (Y, 11*). Let U E 11. There exists a U* E 11* with U ::: U*nX. By Theorem 2.5, there exists a uniformly continuous f:X ~ (M, d) where d is a metric for M, such that for each x E X, f(S(x, U*)) ::: S(f(x), 1/2). Since M is paracompact, the covering V ::: {SCm, 1/2) I m E M} has a locally finite refinement W that belongs to the finest unifonnity for M. Since f is continuous, l (W) E 11* because 11* is fine. Then l (W) IX is a locally finite member or 11 that refines U.
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Similarly, V has a a-disjoint refinement Z. Hence l (Z) is a a-disjoint open refinement of U* that is itself uniform, so l (Z) IX is a a-disjoint unifonn refinement of U.-
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THEOREM 5.13 Each subfine uniform space has a basis of uniformly locally finite coverings and a basis of a-uniformly discrete coverings.
The proof of Theorem 5.13 follows from the preceding lemmas. A unifonnity is said to be point finite if each member has a unifonn point finite refinement. THEOREM 5.14 ffll is a point finite uniformity, so is the derivative Ill. Proof: Let {V anV~} E Ill. Let U::: {Va) and Va::: {V~} for each a. Without loss of generality we may assume U is point finite. Choose a point finite A ::: {A y I E f.! with A <* U and for each a, a W a <* Va in 11. Each A y is contained in at most finitely many of the Va, say Val . . . Van' Put G y ::: W al n . . . nWan and let zy ::: {ZS) be a point finite unifonn refinement of GY. Then {Aynzs) E III is point finite.
To show that {AynZS} <* {VanV~}, note that for any pair 'Y, 0 we can choose Va containing Star(A y , A) and a V~ containing Star('4, wet). If Aenzt
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meets AynZS then A e c Star(A y, A) c U u which implies Ze < G e < W U , so ~ c Star(Z{>" W U ) c vg. Hence Aen~ c AynZS. We conclude that {AynZSI <* {U unV~ I so f..ll is also a point finite uniformity.COROLLARY 5.6 Each sUbfme uniform space is locally fine. Proof: Let (X, f..l) be a subfine space and (Y, f..l*) a fine space such that (X, f..l) is a unifonn subspace of (Y, f..l*). From the definition it is clear that (X, f..ll) is a unifonn subspace of (X, f..l * 1). By Theorem 5.13 f..l and f..l* are point finite, so by Theorem 5.14, f..ll and f..l* I are also. Since f..l* is fine, f..l* 1 = l..l* which implies f..ll = f..l. A straightforward induction on a yields Jlu = f..l for each a, so A,(f..l) = f..l.
-
EXERCISES
DERIVATIVES OF UNIFORMITIES It took over 20 years to settle the question of whether the a th derivative f..lu of a unifonnity f..l is a unifonnity or not. Ginsburg and Isbell announced a proof in 1955 that the answer is affinnative. In their 1959 paper referenced earlier in this section, they retracted their announcement and left the question open. 1. [J. Isbell, 1964] If f..l is a unifonnity that has a basis consisting of point finite coverings, then f..ll (and hence any f..lU) is a unifonnity. 2. [J. Pelant, 1975] A unifonnity f..l for X has a basis consisting of point finite coverings if and only if the first derivative of the product unifonnity I1~1 f..la, where each f..la = f..l, of the product space (X, f..l)K is a unifonnity for each cardinal K. 3. [J. Pelant, 1975] There exists a unifonn space with no basis consisting of point finite coverings. Consequently, by Exercise 2, there exists a unifonnity with a derivative that is not a unifonnity. (Note: The existence of a uniform space without a basis consisting of point finite coverings was also an open problem from Isbell's book.) INJECTIVE SPACES 4. Let X and Y be unifonn spaces and let U(X,Y) denote the collection of all unifonnly continuous functions from X into Y. For an imbedding i:A ~ X, define the function F:U(X,Y) ~ U(A,Y) by F(h) = h © i for each h E U(X,Y). Show that Y is injective if and only if F is onto. 5. Show that if a unifonn space Y has the property that whenever A is a closed
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uniform subspace of X, every unifonnly continuous function J:A ~ Y can be extended to a unifonnly continuous function F:X ~ Y, then Y is complete and hence injective. [Hint: Suppose Y is not complete and Z is its completion. Put A = Y and construct X as follows: Let K be a regular cardinal. K + 1 is a compact Hausdorff space in the order topology. A continuous function Jon K or K + 1 into a metric space is finally (residually) constant. Since Y is imbeddable in a product of metric spaces, K can be chosen large enough to make each continuous function on K finally constant. Also, K should be greater than the cardinality of any subset of Y having a limit point in Z - Y. Let S c Y have a limit point in Z - Y. Put X = [Y X (K + 1) - K]U[S X (K + 1)Ju[P x KJ. Then Xc Z X (K + 1). Identify Y with Y x (K + 1) - K so that Y c X. It can be shown that Y is closed in X and any continuous extension over S x (K + 1) of the identity i: Y ~ Y must agree with coordinate projection on a unifonn neighborhood of S x [(K + 1) - K] and cannot be extended over {p) x K.J 6. Let (X, /-l) be a unifonn space and let p(/-l) be the collection of all coverings of X refined by a finite member of /-lo Show that p(/-l) is a unifonnity for X that is coarser than /-l. 7. Show that a unifonn retract of an injective unifonn space is also injective. A retraction is a mapping r such that r © r = r. A uniform retract (Y, v) of a unifonn space (X, /-l) is a unifonn subspace such that there exists a unifonnly continuous function r:X ~ Y such that r is a retraction.
5.7 Categories and Functors In this section we introduce the elements of category theory. None of the following sections or chapters will depend on a knowledge of this subject, so this section can be skipped without sacrificing the accessibility to subsequent sections. However, there will be exercises at the end of future sections that will involve an elementary understanding of category theory. The approach here is to keep the development of category theory separate from our development of unifonn spaces to allow the reader the choice of pursuing this area or not. Some researchers in the theory of unifonn spaces tend to combine categorical notions with unifonn space concepts in theIr publications. Consequently, a knowledge of category theory is helpful in reading some of the literature in the theory of uniform spaces. On the other hand, the reliance on categorical notions and terminology by these unifonn space researchers has often been criticized as making this field almost inaccessible to anyone but the specialist. In future sections and chapters, all unifonn concepts will be introduced without reliance on categorical terminology or concepts, to allow as immediate access to the ideas as we know how. However, in order to illustrate the current usage of category theory as a tool in the study of unifonn spaces (just as set theory is used as a tool), we will
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present a selection of problems that involve categorical notions that should suffice to prepare the reader to access that branch of uniform space theory that uses category theory. For those who have studied even elementary set theory, it is understood that the class of all sets cannot itself be a set. The assumption that it is leads to a statement that is simultaneously true and false, as first noted by B. Russell. This is the famous Russell Paradox. However, it is still useful to retain the notion of a class of all sets, just it is useful to have the notion of the class of all uniform spaces or the class of all topological spaces. By a concrete category C, we mean a class 0 of sets (referred to as objects of C), and for each ordered pair of objects (X, y), a set of functions f:X -7 Y (denoted by M(X. Y) and called the morphisms of C with domain X and range Y) such that (1) The identity function on each object X is a morphism of C with
domain X and range X. (2) Each composition of morphisms of C is a morphism of C. The class 0 of sets may be larger than any cardinal number, and will be in all examples given in this book. By a covariant functor F:C -7 D of two concrete categories C and D, we mean a pair of functions F 0 (called the object function) and F M (called the morphism function) such that F 0 assigns to each object X in C an object F o (X) in D, and FM assigns to each morphism f:X -7 Y of C a morphism F M(f):F0 (X) -7 F o(Y) such that (3) For each identity morphism Ix of C, F M(1X) = IFo(x). (4) For each composed morphism g © f of C, FM(g © f) = F M(g) © F M(f). Since F 0 only applies to objects and F M only applies to morphisms, it is customary to denote both F 0 and FM simply by F as no confusion is likely to arise. This shorter notation is useful for denoting the composition of covariant functors F:C -7 D and G:D -7 E by G © F(X) = G(F(X) and G © F(f) = G(F(f). An isomorphism is a covariant functor F:C -7 D such that there exists a covariant functor F-1:D -7 C with both F- 1 © F and F © F- 1 being the identity functors on C and D respectively. If the functor F:C -7 D is an isomorphism, the concrete categories C and D are said to be isomorphic and both C and D are said to be instances. of the same abstract category. A categorical property is a property P that if possessed by a concrete category C, is preserved by isomorphisms (i.e., if D is a concrete category isomorphic with C, then D also has property P). Similarly, we can define categorical concepts and definitions.
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For instance, the notion of a mapping f:X ~ Y having an inverse 1 : Y ~ X is a categorical concept in the following sense. If we define f:X ~ Y has an inverse Y ~ X to mean that there exists a mapping y ~ X such that ©f = 1x and f © 1 = 1y, then if X and Y belong to the concrete category C such that F:C ~ D is an isomorphism, then it is easily seen from the definitions that the morphism F(j):F(X) ~ F(Y) also has an inverse.
rl:
rl:
r
r1
On the other hand, the notion that the morphism f:X ~ Y is one-to-one is not a categorical notion. However, being one-to-one implies an important categorical notion that is closely related to being one-to-one. In fact, in uniform spaces, being one-to-one is equivalent to this notion. A morphism f:X ~ Y of a concrete category C is said to be a monomorphism if for each pair of morphisms g:Z ~ X and h:Z ~ X we have f © g = f © h implies g = h. Clearly if f is one-to-one it is a monomorphism. On the other hand, if f is not one-toone then f(x) = fCy) for some x yin X. But then the constant functions g:Z ~ {x} and h:Z ~ {y} are not equal, yet f © g = f © h. Hence f is not a monomorphism. Therefore, f is a monomorphism if and only if it is one-to-one. It is the fact that constant functions are morphisms in the category of uniform spaces that makes monomorphisms equivalent to one-to-one mappings.
'*
Similarly to this left cancellation property that defines the monomorphisms, the right cancellation property defines the epimorphisms. f is said to be an epimorphism if for each pair of morphisms g:Y ~ Z and h:Y ~ Z, we have g © f = h © f implies g = h. Epimorphisms are closely related to the concept of being onto. Indeed, an onto morphism is clearly an epimorphism. But being an epimorphism is a more general property, for iff(x) is merely dense in Y, it suffices to be an epimorphism. To see this, note that if g © f = h © f and y E Y, we can pick a net q> inf(X) converging to y. Since g and h are morphisms (uniformly continuous functions in the category of uniform spaces), g © q> converges to gCy) and h © q> converges to hCy). But g © f= h © fand q> cf(X) implies g © q> = h © q>. Since nets converge to unique limits in uniform spaces, we have gCy) = hCy). Therefore,fis an epimorphism. A morphism f:X ~ X is called a retraction if f2 =f © f =f. If a retraction is either an epimorphism or a monomorphism, it is an identity morphism. By a contravariant functor F:C ~ D of two concrete categories C and D, we mean a functor F that assigns to each object X of C an object F(X), but to each morphism f:x ~ Y of C, a morphism in the opposite direction F(j):F(Y) ~ F(X) of D to C that satisfies property (3) above, but instead of satisfying property (4), it satisfies (4') For each composed morphism g © fof C, F M(g © f) = F M(j) © F M(g). Observe that the composition of two contravariant functors is a covariant func-
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tor. Also, functors of different variances can be composed with the resultant functor being a contravariant functor. A duality is a contravariant functor F:C ---* D such that there exists a contravariant functor F-1:D ---* C with both F- 1 © F and F © F- 1 being identity functors. If C and D are concrete categories, and F:C ---* D is a duality, then every concept, definition, or theorem about C gives rise to a dual concept, definition or theorem about D. To illustrate what is meant by dual concepts and definitions, we consider the concepts of sums and products of objects of an arbitrary concrete category C. As we shall see, these concepts are dual concepts. We have already considered the concepts of sums and products of uniform spaces. We now give categorical definitions for sums and products of objects in arbitrary concrete categories in such a way that the ordinary definitions of sums and products of uniform spaces are equivalent to the categorical definitions in the category of uniform spaces. Let {X a} be a collection of objects in the concrete category C and let {i a) be a collection of morphisms such that i a:X a ---* L for some L E C. Then L is called the sum of the objects {X,,} if the following conditions hold: (5) If f:L ---* Y and g:L ---* Yare distinct morphisms, then for some index a, the morphisms f © i a and g © i a are distinct. (6) For each family of morphisms f a:X a ---* Y, there exists a morphism f: L ---* Y such that f © i a = fa for each a. Similarly, an object rr E C is said to be a product of the objects {X a} if there is a collection {p a} of morphisms such that P a:rr ---* X a for each a, if the following conditions hold: (7) If f:Z ---* rr and g:Z ---* rr are distinct morphisms, then for some index a, the morphism P a © f and P a © g are distinct.
(8) For each family of morphisms f a:Z ---* X a, there exists a morphism f:Z ---* rr such that P a © f = fa for each a. To show that the categorical definitions of sum and product are dual, we must show that if L is the sum of {X a} in C, then rr = F(L) is the product of {F(X a)} in D and vice-versa. For this, assume L is the sum of IX a} in C and put rr = F(L). Suppose F(j) and F(g) are distinct morphisms such that F(j):F(Y) ---* rr and F(g):F(Y) ---* rr. Then f:L ---* Y and g:L ---* Y. Since F- 1 © F is the identity, F- 1 © F(j) = f and [F- 1 © F](g) = g. Therefore, f = g implies [F-l © Fl(j) = [F-I © F](g), so F(j) = F(g) which is a contradiction. Therefore, f and g are distinct. Since L is the sum of {X a}, there exists an a with f © i a ::j:. g © i a. Then by an argument similar to the above, F(f © i a) ::j:. F(g © i a) which implies F(i a) © F(j) ::j:. FU a) © F(g). For each a, put p a = F(i a) and let Z = F(Y). Then if F(j):Z ---* rr and F(g):Z ---* n are distinct morphisms, for some index a, the
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morphisms Pu © F(j) and p u © F(g) are distinct Roughly speaking, condition (5) in C forces condition (7) in D. Next, suppose F(f u):Z --? F(X u) is a family of morphisms in D. Then --? Y is a family of morphisms in C, so by (6), there exists a morphismf:L --? Y with f © i u = fa for each a. But then F(j):Z --? D such that p u © F(j) = F(f u) for each a. Again, roughly speaking, condition (6) in C implies condition (8) in D. Hence D = F(L) is the product of {F(X u») in D.
f u: X u
The proof that if D is the product of (X u) in D, then L = r! (D) is the sum of IF- 1 (X u») is similar. This establishes the duality of the concepts of sum and product in arbitrary concrete categories. To illustrate what is meant by dual theorems we will show that the categorical theorem: sums are unique (in C), gives rise to the dual (categorical) theorem: products are unique (in D). To see this, let L be the sum of IXu) in C. Then D = F(L) is the product of I F(X u») in D as we have already seen. If D' is another product of IF(X u»), then L' = F- 1 (il,) is another sum of IX u) so there exists a uniform homeomorphism i:L' --? L (since sums are unique in C). Therefore, F(i):D --? D' and since i-I © i = IE' and i © i-I = IE, we have:
Similarly, F(i- l ) © F(i) = IF(E), so F(i) is a uniform homeomorphism of D = F(L) onto D' = F(L'). Therefore, products are unique in D. The theorem that products are unique in C implies sums are unique in D is proved similarly. This establishes the duality of these two theorems. The principle of duality says that any categorical concept, definition or theorem gives rise to a dual concept, definition or theorem. It is also of interest to give categorical definitions of the concepts of subspace and quotient space. For the concept of a subspace, we recall that for a uniform space X, a subspace can be considered to be a uniform space S such that there exists a one-to-one uniformly continuous function i:S --? X such that the uniformity of X relativized to i(S) is identical with the identification uniformity of i(S). By a uniform imbedding of S into X, we mean a uniform homeomorphism of S onto a subspace S' of X. Clearly, if S is a subspace of X, the mapping i:S --? X above is a uniform imbedding.
Then a one-to-one mapping i:S --? X is a uniform imbedding if and only if whenever i = g © f where g and f are uniformly continuous and f is one-to-one and onto, then f is a uniform homeomorphism. To see this first assume i:S --? X is a uniform imbedding and i = g © f where g and f are uniformly continuous and f is one-to-one and onto. Let v be the uniformity of Sand Il the uniformity of X restricted to i(S). For V E v, i(V) = {i(V) I V E V} E Il since i is a uniform imbedding. Let A be the uniformity of S' = f(S). Since g is uniformly
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continuous, g-I(i(V» E A. Butf= g-I © i because i is one-to-one and g = i © i which implies g is one-to-one. Hence f(V) E A, so f is a uniform homeomorphism.
r
Conversely, if the one-to-one mapping i = g © f where g and f are uniformly continuous and f is one-to-one and onto implies f is a uniform homeomorphism, then if V E v, f(V) E A. Assume i(V) does not belong to /.!. Then g(g-I U(V))) is not in /.! since g is one-to-one. But f(V) E A implies g -I U(V» E A since f = g-I © i. Therefore, the identification uniformity /.!* of g is strictly finer than /.!. Let A' be the uniformity induced on 5' by /.! with respect to g. Then A' is strictly coarser than A = gl (/.!*). Hence f is uniformly continUOUS with respect to v and A', but (5,v) and (5', A') are not homeomorphic which is a contradiction. Therefore. dV) E /.! so i is a uniform imbedding. Next. we use the characterization of uniform imbeddings to motivate the categorical definition of imbeddings. A monomorphism i:S --7 X in a concrete category C is said to be an imbedding if whenever i = g © f where g and f are morphisms, and f is one-to-one and onto. then f is an isomorphism. This means that if f:5 --7 5' then 5' has the same structure as 5. so that no structure can be inserted on 5 that is between the structure of 5 and that of i(5). In the case of uniform spaces, this means that the uniformity of 5 is the one induced by the function i. It is also clear that in any concrete category, two imbeddings with the same image have isomorphic domains. The dual categorical concept of an imbedding is that of a quotient morphism. An epimorphism q:X --7 Q in a concrete category D is said to be a quotient if whenever q = d © h where d and h are morphisms and d is one-toone and onto. then d is an isomorphism. To show that in the category of uniform spaces, a uniform quotient mapping satisfies this categorical definition, let (X, j..l) be a uniform space and q:X --7 Ya uniform quotient mapping. Then q is onto and Y has the identification unifonnity with respect to /.! generated by q. Let v be the uniformity of Y, then V E v if and only if l (V) E /.!.
r
Let q = d © h where d and h are uniformly continuous and d is one-to-one and onto. Let A be the uniformity of Q' = heX). If W E A, then h -I (W) E /.!. Since q is a uniform quotient, q(h- I (W» E v. But then d(h(h- I (W))) = deW) E v, so d is a uniform homeomorphism. Consequently, in the category of uniform spaces, unifonn quotient mappings satisfy the categorical definition of a quotient morphism. A subcategory S of a concrete category A is a category such that each object of S is an object of A and each morphism of 5 is a morphism of A. S is said to be a full subcategory of A if every morphism of A whose domain and range are both objects of S is a morphism of 5. A full subcategory R of a category A is said to be a reflection of A if for each object X E A, there is an object X* E R and a morphism r:X --7 X* (called the reflection morphism) such that each morphism f:X --7 Y where Y E R can be factored uniquely over X* by r
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(i.e., f = g © r for a unique g:X* -+ Y). The space X* is called the reflection of X in R or simply the R-reflection of X. There are many interesting reflections of the category of all uniform spaces. We have already encountered some of them unknowingly. For instance, the class of all complete uniform spaces C is a reflection of the category U of all uniform spaces. For each uniform space X E U, the completion X* E C is the C-reflection of X and the natural imbedding i:X -+ X* is the reflection morphism such that any morphism f:X -+ Y E C factors uniquely over X* by i. For any reflection R of a category A, an R-reflection of an object X E A is unique up to isomorphism. To see this, note that if X* and X+ are both R-reflections of X, then there exists reflection morphisms r*:X -+ X* and r+:X -+ X+ such that any morphism f:X -+ Y E R factor uniquely over both X* and X+ by r* and r+ respectively. Therefore, r* = g+ © r+ and r+ = g* © r* for some unique g* and g+. Now g*:X -+ X+ and g+:X+ -+ X*. Moreover, r+ = g* © (g+ © r+) = (g* © g+) © r+ so (g* © g+) = lx+. Similarly, g+ © g* = Ix'. Therefore, g + = g*-l so X* and X+ are isomorphic (in the class of uniform spaces this means uniformly homeomorphic). The fact that an R-reflection is unique allows us to make the simplifying assumption that if X E R, the only R-reflection of X is X itself and the only reflection morphism r:X -+ X is the identity morphism. This allows us to state the following fundamental theorem about reflective categories.
THEOREM 5.15 If R is a reflection of a concrete category A, X, YEA and X*, y* are their R-reflections respectively, then any morphism f:X -+ Y determines a unique morphism f* :X* -+ y* such that r © f = f* © s where r:X -+ X* and s:Y -+ y* are the reflections morphisms respectively. Proof: Since s © f:X -+ Y*, it can be factored uniquely over X* by r. Letf* © r be this factorization. Then s © f = f* © r and since f* is unique, it is the unique morphism that satisfies the theorem. Theorem 5.15 shows that the correspondences X -+ X* and f -+ f* determine a functor P called the reflection functor. The object function of P is Po:A -+ R defined by Po (X) = X* for each X E A and the morphism function PM is defined by PM(f) = f* for each f E A. To see that P is a functor, we need to show that Po and PM satisfy (3) and (4) of the definition of a functor. To demonstrate (3) we note that by Theorem 5.15, the identity Ix has a corresponding unique morphism l~:X* -+ X* such that l~ © r = r © Ix = r. Now Ix> © r = r so Ix> = 1~ since G is unique. Therefore, PM(1X) = Ipo(x), To demonstrate (4), if g © f is a morphism of A, then by Theorem 5.15, there exists unique morphisms g*, f* and (g © j)* such that (g © j)* © r =r © (g © j),f* © r = r © fand g* © r = r © g. But then r © (g © j) = g* © r © i=
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g* © 1* © r. Since (g © j)* is unique, (g © j)* = g* © 1*. Therefore, PM(g) © PM(/)'
The dual concept to the concept of a reflection is the concept of a corejlection defined as follows. A full subcategory K of a concrete category A is said to be a coreflection of A if for each object X E A, there exists an object X* E K and a coreflection morphism k:X* ~ X such that each morphism /:Z ~ X where Z E K can be factored uniquely over X* by k (i.e., / = k © g for a unique morphism g:Z ~ X*). The space X* is called the coreflection of X in K or simply the K-reflection of X. As with reflections, there are many interesting coreflections of the category of uniform spaces. Possibly the simplest and most useful is the fine corejlection that maps each uniform space (X, J..l) onto (X, u) where u is the finest uniformity for X. Here, the coreflection morphisms are the identity mappings. To see that the class F of fine uniform spaces is a coreflection of the category U of uniform spaces, it suffices to show that each morphism j:Z ~ X where Z is a fine space can be factored uniquely over (X, u). But since /:Z ~ X is continuous and Z is fine, j:Z ~ (X, u) is uniformly continuous, so / can be factored over (X, u) by the identity mapping. By the principle of duality, for any coreflection K of a category A, a K-coreflection X* of an object X is unique up to isomorphism. As with reflections, the uniqueness of K-coreflections allows us to make the assumption that if X E K, the only k-coreflection of X is X itself and the only coreflective morphism k:X ~ X is the identity morphism, so we have the dual theorem: THEOREM 5.16 1/ K is a corejlection 0/ a concrete category A, X, Y E A and X*, y* are their K-corejlections respectively, then any morphism/:X ~ Y determines a unique morphism 1* :X* ~ y* such that / © k = j © 1* where k:X* ~ X and j:Y* ~ Yare the corejlective morphisms respectively.
Theorem 5.16 shows that the correspondences X ~ X* and / ~ 1* determine a functor K called the coreflection functor. The object function of K is !Co:A ~ K defined by !Co (X) = X* and the morphism function KM is defined by KM(/) = 1* for each/E A.
EXERCISES 1. Show that the definitions of uniform sums and products are equivalent to the categorical definitions of sums and products in the concrete category of uniform spaces.
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DUAL THEOREMS 2. Show that the categorical theorem P: A retraction f that is a monomorphism is an identity implies the dual theorem Q: A retraction f that is an epimorphism is an identity. REFLECTIONS 3. Let U be the category of uniform spaces and for each (X, Jl) E U let p(Jl) denote the uniformity consisting of all coverings refined by finite members of Jl (see Exercise 6, Section 5.6). Then (X, p(Jl» is a precompact uniform space. Show that the class P of all precompact uniform spaces is a reflection of U with reflection morphisms being the identity functions. Let P denote the reflection functor (called the precompact reflection). Show that for each (X, Jl) E U that Po(X, Jl) = (X, p(Jl» and for each morphism f:(X, Jl) ~ (y, v) that PM(j) =f. 4. For each (X, Jl) E U let e(Jl) be the collection of all coverings of X refined by a countable member of Jl. Show that e(Jl) is a separable uniformity for X. [A uniform space is separable if it has a basis consisting of countable coverings.] Show that the class U w of all separable uniform spaces is a reflection of uniform with reflection morphisms being the identity functions. Let e denote the reflection functor (known as the separable reflection), and show that for each (X, Jl) E U, eo(X, Jl) = (X, e(Jl» and for each morphism j:(X, Jl) ~ (Y, v) that eM(j) = f·
5. Show that the separable uniformities are precisely the countably bounded uniformities. 6. Show that the reflections P and e commute, i.e., for each (X, Jl) p(e(X,Jl» = e(p(X,Jl». 7. Let 1t denote the completion reflection that maps each (X, Jl) Show that the reflections P and 1t commute.
E
E
U,
U into /lX.
8. It was long an unsolved problem (due to Ginsberg and Isbell) whether the reflections e and 1t commute. 1. Pelant (1974) showed that in general they do not. THE SAMUEL COMPACTIFICATION 9. Show that the composition 1t © P of the reflection 1t and P is again a reflection. Since for each X E U, p(X) is precompact, it is clear that 1t(p(X» is compact and p(X) is uniformly homeomorphic with a dense subspace of 1t(p(X». 1t(P(X» is called the Samuel compactification of X, or the uniform compactification of X.
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COREFLECTIONS 10. Let U be the category of unifonn spaces and for each X = (X, ~) E U let denote the locally fine coreftection of~. Put X* = (X, A(~» and let L be the category of locally fine unifonn spaces. Show that A:U ~ L defined by A(X) = X* for each X E U and A(j) = f is a coreflective functor so that the locally fine coreflection is really a coreftection categorically. A(~)
Chapter 6
PARACOMPACTIFICATIONS
6.1 Introduction In the late 1950s and during the 1960s, K. Morita and H. Tamano worked (independently) on a number of problems that involved the completions of uniform spaces. We state some of these problems here even though we have not yet defined some of the terms used in the statements of these problems. (1) (Tamano) Characterize the uniform spaces with paracompact completions" (2) (Morita) Characterize the Tychonoff spaces with paracompact topological completions. (3) (Morita) Characterize the Tychonoff spaces with LindelOf topological completions. (4) (Morita) Characterize the Tychonoff spaces with locally compact topological completions. (5) (Tamano) Is there a paracompactification of a Tychonoff space analogous to the Stone-Cech compactification or the Hewitt realcompactification? By the topological completion of a uniform space, we mean the completion of the space with respect to its finest uniformity. By a paracompactification of a topological space, we mean a paracompact Hausdorff space Y such that X is homeomorphic with a dense subspace of Y. If we identify X with this dense subspace (which is customary), then Y is a paracompact Hausdorff space containing X as a dense subspace. Certain paracompactifications are compact. We call these compactifications. The theory of compactifications is well developed. The theory of paracompactifications is not. Notable among compactifications is the Stone-Cech compactification. We will devote a significant amount of effort in this chapter to its development. We are not yet in a position to define what a rea[compact space is. Like paracompactness, realcompactness is a generalization of compactness. We will show in the next chapter thatfor all practical purposes, realcompactness is also a generalization of paracompactness. We will of course, formalize what we mean by "for all practical purposes," but for now, we only mention that this formalization is a set theoretic matter. It turns out that the statement All
6.1 Introduction
157
paracompact spaces are realcompact is known to be consistent with the axioms of ZFC, but it is not known if it is independent of ZFC. Furthermore. it is known that if there exists a Tychonoff space that is paracompact but not realcompact, that its cardinality is larger than any cardinal with which we are familiar.
Once we know what a realcompact space is, we will be able to define a rea/compactification analogously with the way we defined a paracompactification. Like the theory of compactifications, the theory of realcompactifications is well developed. Among the various realcompactifications a space may have, there is one, known as the Hewitt realcompactification that plays a role in the theory of realcompactifications analogous to the role played by the Stone-Cech compactification in the theory of compactifications. Since "for all practical purposes," paracompactifications lie between realcompactifications and compactifications, it is natural to ask if there exists a paracompactification that plays a role analogous to the role played by the Stone-Cech compactification and the Hewitt realcompactification in the theories of compactifications and realcompactifications respectively. This is precisely Problem 5 above that we will refer to as Tamano's Paracompactification Problem. In this chapter we will show that, in general, there does not exist such a paracompactification. We will also characterize those spaces for which such a paracompactification exists and examine some of their properties. In 1937, two papers appeared that characterized the Tychonoff (uniformizable) spaces as those having compactifications. The two papers were: Application of the Theory of Boolean Rings to General Topology that appeared in the Transactions of the American Mathematical Society (Volume 41, pp. 375-481) by M. H. Stone and On bicompact spaces by E. Cech that appeared in Annals of Mathematics (Volume 38, pp. 823-844). Their approaches were very different, but both exhibited a compactification that is the largest possible compactification of a Tychonoff space. Clearly every compactification of a Tychonoff space X is the completion of X with respect to some totally bounded uniformity (since we can relativize a compactification's unique uniformity to X to obtain the desired totally bounded uniformity on X). In 1948, P. Samuel published a study of compactifications obtained as completions of uniform spaces (see Ultrafilters and compactifications, Transactions of the American Mathematical Society, Volume 64, pp 100-132). As a result, compactifications obtained as the completion of a uniform space are sometimes called the Samuel compactification of the uniform space. In this chapter we will employ this approach. The Stone-Cech compactification will be obtained as the completion of a Tychonoff space with respect to its ~ uniformity (introduced in Chapter 3). If X is uniformizable, the
6. Paracompactifications
158
completion (X',W) of (X,~) is usually denoted by central role in general topology.
~X.
The study of
~X
plays a
Since ~ is the finest totally bounded uniformity, the Stone-Cech compactification is realizable as the completion with respect to the finest uniformity for which a compactification exists. A similar situation occurs with respect to realcompactifications. We will find in the next chapter that realcompactifications are precisely the completions with respect to the countably bounded uniformities and the Hewitt realcompactification is the completion with respect to the finest countably bounded uniformity (namely the e uniformity). Consequently, if Tamano's Paracompactification Problem is to be answered affirmatively for a given space X, the solution needs to be the completion of X with respect to the finest uniformity for X that has a paracompact completion. K. Morita was interested in when the topological completion of a Tychonoff space is paracompact. He showed that if X is an M-space (to be defined later in this chapter), then X has this property. Morita called the topological completion of an M-space X the "paracompactification of X." Consequently, we will call a paracompactification that is a solution to Tamano's Paracompactification Problem the Tamano-Morita paracompactification. In studying Tamano's Paracompactification Problem, several questions come to mind. The first is: Which spaces admit a paracompactification? This question is easily answered (for T J spaces). Since compactifications are paracompactifications, the existence of the Stone-Cech compactification implies that all Tychonoff spaces have paracompactifications. Conversely, if a space has a paracompactification it must be Tychonoff since paracompact Hausdorff spaces are normal (hence uniformizable) and as we saw in Chapter 5, a subspace of a uniform space is also a uniform space. Consequently, the class of spaces is not expanded by considering those with paracompactifications as opposed to those with compactifications. The next question that arises is: Are all paracompactifications obtainable as a completion with respect to some uniformity? This question is also easy to answer. If X is a Tychonoff space and PX some paracompactification of X, then X is a dense subspace of PX. Since PX is paracompact, by Theorem 4.4, it is cofinally complete with respect to its finest uniformity u, and hence complete with respect to u. Now u induces the v (derived) uniformity on X (see Section 4.5). Since completions are unique (Theorem 4.10), (PX,u) is the completion of (X,v). Consequently, all paracompactifications are obtainable as completions with respect to some uniformity. A completion of a uniform space that is cofinally complete will be called a uniform paracompactification. We have already seen (Theorem 4.12) a necessary and sufficient condition for a uniform space to have a uniform paracompactification. Clearly, a Tychonoff space can have many paracompact_ ifications in the same manner that it can have multiple distinct Samuel
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159
compactifications. We begin our study of paracompactifications by first considering compactifications. We wish to obtain the Stone-Cech compactification of PX of a Tychonoff space X as the completion of X with respect to its P uniformity. Since Phas a basis A consisting of all finite normal coverings it is clearly precompact, so the completion of eX, P) is compact. But what makes the completion of eX, P) the Stone-Cech compactification? E. Cech characterized PX in the following way:
THEOREM 6.1 (E. Cech, 1937) Any Hausdorff compactification BX of X is the image of PX under a unique continuous mapping f that keeps X pointwise fixed and such that f(~X - X) = BX - X. M. H. Stone characterized ~X in another way:
THEOREM 6.2 (M. H. Stone. 1937) Iff is any continuous mapping of a Tychonoff space X into a compact Hausdorff space Y. then f has a unique continuous extension ff'J :~X -7 Y. We propose to show that the completion of (X, ~) satisfies Theorems 6.1 and 6.2 in the sense that if we replace ~X with the completion of (X, ~) in the statement of both these theorems, they both remain true. Then we will show that any compactification satisfying Theorems 6.1 and 6.2 is the completion of eX, P). Thus, among the various compactifications a Tychonoff space X may have, the completion of eX, P) is distinguished as the Stone-Cech compactification of X.
6.2 Compactifications In what follows, we will adopt the following notation: if (X, f..l) is a uniform space, then f..lX will denote its completion. For us then, ~X will denote the completion of (X, ~). We first prove Theorem 6.2 using this interpretation of ~X.
(Proof of Theorem 6.2) Let Wbe the unique uniformity of ~X (Theorem 2.8). Then (BX, W) is the completion of (X, ~). Let B be the unique uniformity on Y. By Lemma 3.8, each Tychonoff space admits a uniformity that has a basis consisting of all finite normal coverings. Let U E B be one of them. For each UE UputVu=Unj(X)andlet V= (VUIUE Ul. Then V is a finite normal covering of I(X) and 1 (V) = {f-l (Vu) IU E U} is a finite normal covering of X. By definition of B, r 1 (V) E B. But then f is uniformly continuous with respect to ~ and B. By Proposition 4.20, Y is complete, so by Theorem 4.9, I has a unique uniformly continuous extension/f'J:BX -7 Y.-
r
(Proof of Theorem 6.1) X can be identified with dense subspaces of both BX and BX. Define f:X c ~X -7 X c BX by I(x) = x for each x E X. Then, as
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in the proof of Theorem 6.2, B' (the unique uniformity on BX) has a basis consisting of all finite normal coverings. If U' E B' is one of these finite normal coverings, we have just seen that r 1 (Unf(X)) is a finite normal covering of X, so f is uniformly continuous. Therefore, we can apply Theorem 6.2 and obtain a unique continuous extension f~:PX ~ BX. Since PX is compact, f(PX) is closed in BX. But X c f(PX) c BX which implies f(PX) = BX since CI(X) = BX. It remains to show that f~(PX - X) c BX - X. For this let x E PX - X and let {x u} C X be the fundamental Cauchy net in X (see Theorem 4.9) that is used to define x' = j'(x). Suppose x' E X. Now {xu} converges to x and {f(x u )} converges to x'. But since f(x) = x for each x E X, {f(x u)} = {x u}. But then {xu} converges to x' E X and {xu} converges to x which is not in X which is a contradiction. Hence x' does not belong to X. Therefore,j~(pX - X) c BX - X. Since f~(X) = X, we have f~(PX - X) = BX - X.Finally, assume BX is a compactification of X that satisfies Theorem 6.1. Then there exists a unique uniformly continuous function f:BX ~ PX such that f(BX) = PX, f(x) = x for each x E X, and f(BX - X) = PX - X. Let U E P be a finite normal covering of X. Then there exists a finite normal covering U' of W such that U' nX = U. Since f is uniformly continuous,j-l (U') is a finite normal covering of BX that belongs to B'. But then r 1(U')nX E B. Since f(x) = x for each x E X and f(BX - X) = PX - X, r1(U')nX = r1(U'nX) = rl(U) = U. Hence B contains all finite normal coverings of X, so pcB. But by Lemma 3.8 and Theorem 2.8, the unique uniformity B' on BX has a basis consisting of all finite normal coverings (on BX). Consequently B has a basis consisting of (perhaps not all) finite normal coverings so B c p. But then B = p, so BX is pX. In order to demonstrate the utility and importance of the Stone-Cech compactification, this section and the next two will be devoted to an analysis of some of the most important topological properties of Tychonoff spaces in terms of subsets of pX. The approach we will follow is due to H. Tamano, and appeared in his 1962 paper titled On compactifications which appeared in the Journal of Mathematics of Kyoto University (Volume I, Number 2). Tamano's approach is usually thought of as characterizing topological properties of a Tychonoff space X in terms of the behavior of subsets of PX - X (which it does). But as we shall see, his approach is deeply involved with certain normal coverings and with extending uniformities of X into pX. In this section we build the tools needed for this development. In the next section we prove Tamano's Completeness Theorem, and in Section 6.4, we present Tamano's Theorem. In Chapter 2, we defined an open set A c X to be regularly open if intx(Clx(A)) = A. If in turn, X is a subspace of Y, we can generalize this concept by taking the closure and the interior in Y rather than in X in order to get what is called an extension of A over Y. In general, if U is open in X and D' is open in Y such that V = V'nX, then V' is said to be an extension ofD over Y. Put VE(y) = Y - Cly(X - V). Then vEry) is an extension of Dover Y since
6.2 Compactifications
161
UE(Y)nX = (Y - Cly(X - U»nX = X - Cly(X - U)nX = X - (X - U) = U. We call UE(Y) the proper extension ofU over Y. It is clear from this definition that U c V implies UE(Y) e VEly). LEMMA 6.1 Let X be a dense subspace of Y and let U be open in X. Then UElY) is the largest extension of U over Y. Proof: Let U' be an extension of U over Y. If Y does not belong to UE(Y), then y E Cly(X - U). Therefore, every open set V'(y) c Y containing y meets X - U which implies V'(y)nX is not a subset of U = U' nX. Consequently, V'(y) is not contained in U' which implies y does not belong to U' since U' is open. Therefore, U' c UE(y). • LEMMA 6.2 Let X be a dense subspace of Y. Then for any A eX. we have Intx(Clx(A)) = Inty(Cly(A))nX. Proof: Let x E Intx(Clx(A». Then there exists an open set U(x) containing x such that U(x) c Clx(A) c Cly(A). But then there exists an open set U'(x) c Y such that U'(x)nX = U(x). Suppose U'(x) is not a subset of Cly(A). Then there exists ayE U(x) with y E Y - Cly(A) which is open. Since X is dense in Y, U'(x)n[Y - Cly(A)]nX oF 0. Let Z E U'(x)n[Y - Cly(A)]nX which implies Z E U'(x)nX = U(x) c Cly(A). But on the other hand, Z E Y - Cly(A) which implies Z does not belong to Cly(A) which is a contradiction. Therefore, U'(x) c Cly(A) which implies x E Inty(Cly(A»nX so Intx(Clx(A» c Inty(Cly(A»nX.
Conversely, if x E Inty(Cly(A»nX then x E X and there exists an open set U'(x) e Y containing x such that U'(x) c Cly(A) which implies U'(x)nX c Cly(A)nX. Put U(x) = U'(x)nX. Then U(x) is an open set in X containing x with U(x) e Cly(A)nX. Suppose y E Cly(A)nX. Then every open set V'(y) c Y containing y meets A. But then V'(y)nX meets A. Since A c X, every open set V(y) c X containing y meets A which implies y E Clx(A). Therefore, Cly(A)nX e Clx(A). Hence U(x) c Cly(A)nX c Clx(A). Therefore, x E Intx(Clx(A», so Inty(Cly(A»nX e Intx(Clx(A». • LEMMA 6.3 Let X be a dense subspace of Y. If u' is an extension of the open set U eX. then Cly(U') = Cly(U). Proof: Since U = U'nX we have U c U' which implies Cly(U) c Cly(U'). Suppose y E Cly(U'). Then every open set V'(y) c Y containing y contains a point of U'. Since V'(y)nU' oF 0 and since X is dense in Y, there exists an x E X such that x E V'(y)nU' which implies x E U. Therefore, every open set V'(y) c Y containing y meets U which implies y E Cly(U). Hence Cly(U') c Cly(U) so Cly(U,) Cly(U) .•
=
162
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LEMMA 6.4 Let X be a dense subspace of Y. Then VE(y) is regularly open in Y if and only if V is regularly open in X. Also, if V is regularly open in X, then VE(Y) = Inty(Cly(V)). Proof: If VE(Y) is regularly open, then V = VE(Y)nX = Inty(Cly(VE(Y))nX. Since VE(Y) is an extension of V, by Lemma 6.3 we have V = Inty(Cly(V»nX. Then by Lemma 6.2, V = Intx(Clx(V» so V is regularly open. Conversely, suppose V is regularly open. Then V = Intx(Clx(V» = Inty(Cly(U»nX. Hence V' = Inty(Cly(U» is a regularly open extension of V over Y. Therefore, V' c VE(Y) by Lemma 6.1. We can complete the proof by showing VE(Y) c V'. For this, we suppose on the contrary that VE(Y) is not a subset of V' which implies VE(y) is not a subset of Cly(U'). Since X is dense in Y we have [VE(Y)n(Y Cly(V,)]nX"# 0. Therefore, V = V£(Y)nX is not a subset of Cly(V') which is a contradiction. Hence VE(Y) c V' so VE(Y) = Inty(Cly(U».PROPOSITION 6.1 Let X be a dense subspace of Y and Y a dense subspace of Z. Let V and V be open sets in X and Y respectively. Then the following are valid: (1) VE(Z) nY = VE(Y) , (2) [VE(y)]E(Z) = VE(Z) and (3) VE(Z) c [V nX ]E(Z). If V is regularly open then VE(Z) = [V nX ]E(Z). Proof: To prove (1) we note that VE(Z)nY = [Z - Clz(X - V)]nY = Y [Clz(X-V)nY] = Y - Cly(X - V) = VE(Y). To prove (2) observe that [VE(y)]E(Z) = Z-Clz(Y - VE(Y) = Z - Clz[Y - (Y - Cly(X - V»] = Z - Clz(Cly(X - U» = Z-Clz(X-U) = VE(Z). To prove (3), note that V is an extension of vnX so by Lemma 6.1, V c (V nX)E(Y) and hence VE(Z) c [V nX ]E(Z) by (2) above. If V is regularly open, then vnX is also regularly open since vnX = Inty(Cly(V»nX = Inty(Cly(VnV»nX by Lemma 6.3. But then by Lemma 6.2, vnX = Intx(Clx(VnX». Therefore, V = (V nX)E(Y) by Lemma 6.4. Again by (2) above, VE(Z) = (V nX)E(Z) . _
If we take an open covering U of X and then take the proper extension of each member of U to BX, we get a new covering W(PX) of X in BX called the proper extension of U into BX and we call the set uUE(PX), denoted E u, the extent of U in BX. An open covering V of X is said to be stable if there exists a normal sequence {V n} of open coverings of X such that VI < V and E v = E Vn for each n. The first of Tamano's theorems that we prove shows that an open covering is stable if and only if its extent is paracompact. In general, if X is a dense subspace of Y, we can define the proper extension VE(Y) of V over Y and the extent uVE(Y) of V in Yanalogously.
Letf:X -+ R (reals). We denote by Z(j) the zero set of fdefined by Z(j) = X If(x) = O} and by O(j), the complement of Z(j) in X. We denote by Z(X) the set of all zero sets of X. In Section 1.1, O(j) was called the support of f. Also recall from Section 1.1 that a partition of unity on a space X is a family {x
E
6.2 Compactifications
163
= {A} of continuous functions ,,:X ~ [0,1] such that 11..,,(x) = 1 for each x E X, and such that all but a finite number of members of
THEOREM 6.3 (J. Dieudonne, 1944) For each locally finite open covering {Va} of a normal space, there exists a partition of unity {a} subordinate to {Va} such that O(a) C Vafor each a. Proof: Let X be a normal space and {Va} a locally finite open covering of X. By Lemma 1.3, {Va} is shrinkable to an open covering {Va}. By Theorem 0.3, for each a there exists a real valued continuous function f a:X ~ [0,11 with fa(CI(Va» = 1 andfa(X - Va) = O. Since {Va} is locally finite, so is IVa}. Hence for each p E X,fa(P):t 0 for at most finitely many a. Thus F:X ~ [OJ) defined by F(x) = La!a (x) for each x E X is well defined. To show F is continuous, let p E X. Then there exists an open W(P) containing p such that W(P) meets only finitely many Va, say V aj • • . Van' Therefore, F(x) = L7=da,(x) for each x E W(P). Since each fa, is continuous on W(P), so is F. But then F is continuous at each point of X. For each a, define
PROPOSITION 6.2 A regular Linde16f space has the star-finite property. Proof: Let X be a regular Linde16f space and let U be an open covering of X. For each p E X, there exists an open set V(P) such that CI(V(P» c U for some U E U. Then V = {v(p)lp E X} is an open covering of X and as such has a countable subcovering say {V(pn)}' For each positive integer n let V n E U such that CI(V(pn» c Vn. Since X is normal by Lemmas 1.2 and 3.6, there exist real valued continuous functions fn on X such that fn(CI(V(pn))) = 0 and fn(X-Vn) = 1, so we can apply Theorem 2.16 which yields a star-finite
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refinement of U. Consequently, a regular LindelOf space has the star-finite property. PROPOSITION 6.3 If X is a regular space and there exists an open covering U of X such that each member of U is LindelOf and U admits a starfinite refinement V, then X can be decomposed into a sum of disjoint open sets, each of which is Lin de IOf and X isfully normal and has the star-finite property. Proof: If x,y E X and there exists a positive integer n with y E Starn(x, V) then we put x - y. As was seen in Section 2.4, this relation can be used to decompose X into disjoint sets X a, a E A, such that x and y belong to the same X a if and only if x - y [see Theorem 2.15]. Then for each a E A, Starn(x, V) c X a for any x E X a' Since X a N ~ = 0 if a '# ~,X a is both open and closed.
To show a particular X a is LindelOf, we first show that for any open covering W of X and any pair (x,n) such that x E X a and n a positive integer, that there exists a countable subset of W covering Starn(x, V). We induct on n. First note that Star (x, V) c U for some U E U and U is LindelOf. Next assume Starn(x, V) has this property. Then there exists a countable collection {Vi} c V that covers Starn(x, V) which implies Star n+ 1 (x, V) C U;=I Star (Vi, V). But Star (Vi, V) C Vi for some Vi E U, so Starn +1 (x, V) C U;=I Vi' Since each Vi is LindelOf, Star n+ 1 (x, V) can be covered by countably many members of W. But then, for any open covering W of X, X a = U;;'=I Starn(x, V) can be covered by countably many members of W. Since X a is closed, X a is LindelOf. This proves the first part of the proposition. The second part follows from Lemma 3.6 and Proposition 6.2. A space X is said to be locally compact if each p compact neighborhood.
E
X is contained in a
LEMMA 6.5 A locally compact Hausdorff space is completely regular. Proof: Let X be a locally compact space and let p be a point of X contained in some open set W. Since X is locally compact, there exists an open set V containing p such that CI(V) is compact. Put U = VnW. Then CI(U) is also a compact neighborhood of p. Since CI(U) is a compact Hausdorff space, it is normal, so there exists a continuous function g:CI(U) ~ [0,1] such that g(P) = 1 and g(CI(U) - U) = O. Let h be the constant function on X - U defined by h(x) = o for each x E X - U. Define f on X by f(x) = g(x) if x E U or f(x) = h(x) otherwise. Clearly f:X ~ [0,1] is continuous, f(P) = 1 and f(X - W) = O. Consequently X is completely regular.LEMMA 6.6 If U is an open covering of a Hausdorff space X such that the closure of each member of U is compact, and if U admits a star-finite refinement V, then X isfully normal and has the star-finite property.
165
6.2 Compactifications
Proof: Clearly X is locally compact so by Lemma 6.5, X is completely regular. Moreover, any set U of U has the property that for any open covering W of X, U can be covered by countably many members of W. This was the only use made of the Lindelbf property in Proposition 6.3. Hence the same method of proof can be used to prove this lemma. THEOREM 6.4 (K. Morita, 1948) A necessary and sufficient condition for a locally compact Hausdorff space to be paracompact is that it possess the star-finite property. Proof: Let X be a locally compact Hausdorff space. Assume first that X possesses the star-finite property. Then each open covering has a star-finite refinement. Since a star-finite refinement is locally finite, we conclude that X is paracompact. Conversely assume X is paracompact. By Stone's Theorem (l.l), X is fully normal. Since X is locally compact, there exists an open covering U such that the closure of each member of U is compact. Since X is fully normal, U has a star refinement. Then by Lemma 6.6, X has the star-finite property. • E. Cech, in his paper On bicompact spaces referenced in Section 6.1, showed that a T 1 space X is normal if and only if Cll3x(E)nCll3x(F) = 0 for each pair of disjoint closed sets E, F c X. This useful result is needed to characterize the locally compact Hausdorff spaces as those spaces X such that BX - X is compact for any compactification BX of X. This result will also be needed in the proof of Taman 0 ' s characterization of stable coverings.
THEOREM 6.5 (E. Cech, 1937) A T] space X is normal if and only if CI 13x (E)nCI 13x (F) = 0 for each pair of disjoint closed sets E, F eX. Proof: Let X be a T 1 space and assume that for disjoint closed sets E, F c X, Cll3x(E)nCll3x(F) = 0. Since pX is normal, there exist disjoint open sets U,V c pX such that Cll3x(E) c U and Cll3x(F) c V. Then UnX and VnX are disjoint open sets in X containing E and F respectively, so X is normal. Conversely, assume X is normal and that E and F are disjoint closed sets in X. Then there exists a real valued continuous function f:X ~ [0,1] such that feE) = 0 and f(F) = 1. By Theorem 6.2, f can be extended to a continuous functionf 13 :pX ~ [0,1]. Since E is dense in Cll3x(E) andf(E) = O,fI3(Cll3x(E» = O. Similarly f 13 (CI 13x (F» = 1. Then U = {x E pX ifl3 (x) < 1/2} and V = {x E PX ifl3 (x) > 1/2} are disjoint open sets containing Cll3x(E) and Cll3x(F) respectively, so CI 13x (E)nCI 13x (F) = 0 .•
PROPOSITION 6.4 A Tychonojf space X is locally compact if and only
if BX - X is compact for any compactification BX of X. Proof: For each x
E
X, there exists a neighborhood U(x) of x such that
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Clx(U(x» is compact, if X is locally compact. Then Clx(U(x» is closed in BX so Clx(U(x)) = Cl ~x(U(x». Therefore. no point of BX - X is contained in Cl ~x(U(x)), so X is open in BX. Hence BX - X is closed and therefore compact. Conversely, if BX - X is compact, then it is closed so for each x E X, there exists a neighhorhood U(x) of y in ~X such that C!(U(x»n[BX - Xl = 0. But then Cl~x(U(x» = Clx(U(x». Hence Clx(U(x» is compact. We conclude that X is locally compact.We are finally in a position to prove the first major theorem from Tamano's 1962 paper. This theorem not only estahlishes the connection among stahle coverings, star-finite partitions of unity and coverings with paracompact extents, hut it is essential in the proof of Tamano's characterization of the LindelOf property in terms of compact subsets of BX - X (where BX is a compactification of X) that we will encounter later.
THEOREM 6.6 (H. Tamano. 1962) The following statements are equivalent for a Tychonojf space x: (1) V is a stable covering ofX. (2) the extent Ev of V in ~X is paracompact and (3) there exists a star-finite partition of unity {
Idx(y) - dAz) I = Id(x,y) - d(x,z) I :;:; Id(x,z) + d(z,y) - d(x,z) I == Id(y,z) I < E.
6.2 Compactifications
167
By Theorem 6.2. the restriction of d x to X can be uniquely extended to a continuous function d~:~X ---7 [0.1]. To show that d~ agrees with d x on Ev we pick Y E Ev - X. and assume drCY) 7: d~(y). Now there exists a net {Ya} eX that converges to Y since d x and d~ are both continuous. {dx(v a )} converges to dAy) and {d~(Ya)} converges to d~(y). But since {Ya} c X, {dxCYa)} = {d~CYa)}. Consequently, {dx(y a)} must converge to two distinct points in a Hausdorff space which is impossible. We conclude that d x can be uniquely extended to ~X. Now d~(r) ~ 1/4 since r E CI~x(U(x». so there exists an open W(r) in ~X containing r such that d~(v) < 1/2 for each Y E W(r)nE v. Therefore, W(r)nE v c sex, 21) = S(x.V)) C Vf for some VE E V'. By Lemma 6.1, W(r) c V' so r E E p. Consequently. Cl ~x(U(x» c E v· Put U = {U(x)lx E Ev}. Since (Ev,'t) is paracompact, there exists a locally finite open refinement {V A} covering E v. Since the coverings V~ consist of open sets in ~X. the set B v = {V~ I V;, E V~ for some n} is a basis for "C. But B v is a subset of the topology on ~X so the topology induced on E v by ~X is finer than 'to Consequently. the covenng {V A} is also open in ~X. An argument similar to the one above can be used to show that Cl ~x(V),J c E v for each A. To show that E v is a paracompact subset of ~X, let {G a} be an open covering of E v. Then for each A, Cl ~x(U A) is covered by finitely many of the Ga's say G I ... Gm • PutH~ = VAnG K for 1(= 1 ... m. Then the family {HO is a locally finite open refinement of {G a}. This completes the proof of (1) ---7 (2). (2) ---7 (3). Assume that E v is paracompact. Since E v is open in ~X, ~X E v is closed and hence compact. Since ~X is also a compactification of E v. by Proposition 6.4, E v is locally compact. Then by Theorem 6.4. E v possesses the star-finite property. Since E v is paracompact. there exists a normal sequence of open coverings {V~} in E v such that VI < Vf • Consider the open covering U = {S(z.V~) IZ E E v}. Since E v has the star-finite property, by Theorem 6.3. there exists a star-finite partition of unity $' = {
It remains to show that E w = E v. For this tirst note that W < V implies Ew c Ev· To show that Ev c Ew, let Z E Ev. Pick x E W'(z)nX, which is possible since X is dense in ~X. Then Ltc I
6. Paracompactifications
168
(3) --7 (1). Let
= Vn
C
V~nX = [V~nX']nX
by Proposition 6.1.(1). Therefore, V~nX' contains V~ so uV~ contains uV~ which implies E vn contains X' for each non-negative integer n. To show that Ev n c X' for each n, let p E Ev n . Then p E V~(x) for some x E X. Let y E
V~(x)nX. Then ~1
L~I I
= L~l
1.
Now ~ I
--7
(1).-
The next step in Tamano's development is to strengthen the hypothesis and conclusion in Theorem 6.6 in the following way: a locally compact Hausdorff space is said to be a-compact if it is the union of countably many compact spaces. In what follows we show that the locally compact paracompact Hausdorff spaces are precisely the topological sums of a-compact spaces. Hence the extents of stable coverings are topological sums of a-compact spaces as can be seen in the proof of (2) --7 (3) in Theorem 6.6. Let us agree to call a
6.2 Compactifications
169
stable covering strongly stable if its extent is (J-compact. Tamano's next theorem establishes the connection between strongly stable coverings and countable star-finite partitions of unity. There are a number of alternate definitions of both local compactness and (J-compactness that are very useful. It will facilitate our development to establish these equivalent deftnitions. The proofs of the next two lemmas are left as exercises (Exercises 1 and 2). LEMMA 6.7 The following four properties are equivalent: (I) X is a locally compact Hausdorff space. (2) For each x E X and neighborhood U of x, there exists a neighborhood V of x with CI(V) c U and CI(V) compact. (3) For each compact K and open U containing K, there exists an open V with K eVe CI(V) c U and CI(V) is compact. (4) X has a basis consisting of open sets with compact closures. LEMMA 6.8 The following three properties are equivalent: (I) X is a (J-compact space. (2) There exists a sequence {Un) of open sets with each CI(Un) compact such that CI(Un) c Un+ 1 for each n, and X = U;;'=l Un. (3) X is a locally compact LindelOf space.
At first glance it might appear that any space that is the union of at most countably many compact spaces is (J-compact, but the rational subset R of R (reals) is a counter example because R is not locally compact. Consequently, local compactness is an essential part of the definition of (J-compactness. Since a (J-compact space is Lindelbf by Lemma 6.8 and completel y regular by Lemma 6.5, then by Lemma 3.6 it is paracompact. Therefore, we get locally compact paracompact spaces by forming topological sums of (J-compact spaces. The next proposition shows that this is the only way to get such spaces. LEMMA 6.9 If a covering {U a}. a E A, has a point (locally) finite refinement {V J3}. ~ E B, then {U a} has a precise point (locally) finite refinement {W a}. If each V J3 is open, then each Wa can be chosen to be open. Proof: Define a function f:B ~ A by assigning to each ~ E B some a E A such that V J3 c U a. For each a let Wa = u{ V J3lf(~) = a}. It is possible that Wa = for some of the a's. Clearly Wac U a for each a, and {W a} is a covering because each V J3 is a subset of some Wa. If {V J3} is point (locally) finite, then each (some neighborhood of) x E X lies in at most finitely many of the V J3 and consequently cannot meet more than finitely many of the W a's. Therefore, {W a } is a point (locally) finite refinement of {U a}. -
o
170
6. Paracompactifications
PROPOSITION 6.5 The locally compact paracompact Hausdorff spaces are precisely the topological sums of a-compact spaces. Proof: In view of the remarks preceding Lemma 6.9, it is clear that we only need to prove that a locally compact, paracompact, Hausdorff space X is a topological sum of a-compact spaces. Let X be covered by open sets U(x) such that x E U(x) and CI(U(x» is compact for each x E X. By Lemma 6.9, there exists a precise locally finite open refinement {V(x) Ix E X I. Since CI(V(x» is compact for each x E X, it meets at most finitely many other V(y)'s. To see this, for each Z E CI(V(x», V(z) meets at most finitely many other V(y)'s and since CI(V(x» is compact, it can be covered by finitely many V(y)'s, say V(z I) ... V(zn). Then W = U?=I V(z;) contains CI(V(x» and meets at most finitely many V(y),s.
For each pair x,y E X put X - Y if there exists a finite family of sets V(z I) .. V(zn) such that x E V(ZI), y E V(Zn) and V(Zi)nV(Zi+d oF 0 for i = 1 ... n-1. Clearly - is an equivalence relation in X. Let {X a I a E A I denote the family of equiValence classes with respect to -. Then the X a cover X and are pairwise disjoint. Each X a is open since it is the union of V(y)'s. Therefore, each X a is a locally compact Hausdorff space. It remains to show that each X a is a-compact. Let K I = CI(V(x» for some x E X a' For each positive integer n, inductively define Kn as the union of all CI(V(y»'s with V(y) meeting K n_l • Since K n - I is compact, it meets at most finitely many V(y)'s (as shown above) so Kn is compact. Since for each y E X a there are finitely many V(z)'s, say V(z d ... V(zn) with x E V(z d, y E V(zn) and V(z;)n V(zi+d oF 0 for each i = I ... n - 1, it is clear that y E K j for some positive integer j. Therefore, X a = U';;=1 KnoB Y Lemma 6.8, X a is a-compact. Therefore, X is a topological sum of a-compact spaces. THEOREM 6.7 (H. Tamano, 1962) The following statements are equivalent for a Tychonoff space x: (I) V is a strongly stable covering of X, (2) there is a countable star-finite partition of unity {
The proof of this theorem can be obtained by modifying the proof of Theorem 6.6 and is left as an exercise (Exercise 3). EXERCISES 1. Prove Lemma 6.7.
2. Prove Lemma 6.8. 3. Prove Theorem 6.7.
6.3 Tamano's Completeness Theorem
171
6.3 Tamano's Completeness Theorem In 1944, in a paper referenced in Section 1.2, J. Dieudonne showed that the product X x Y of a paracompact space X with a compact space Y is paracompact. Hence X x Y is normal. Thereafter, many researchers tried to determine if the reverse implication held; i.e., does the normality of X x Y imply X is paracompact when Y is compact? By 1960, many partial results had been published. Then in 1960, in a paper titled On paracompactness published in the Pacific Journal of Mathematics (Volume 10, pp. 1043-1047), Tam an 0 showed that it does if Y = ~X (more generally if BX is any compactification of X, the normality of X x BX implies that X is paracompact). This result has come to be known as Tamano' s Theorem. In Section 4.5 it was mentioned that H. Tamano and K. Morita had considered the problem: "What is a necessary and sufficient condition for a uniform space to have a paracompact completion?" Tamano published four other papers in 1960. One of them titled Some properties of the Stone-Cech compactijication published in the Journal of the Mathematical Society of Japan (Volume 12, pp. 104-117) details his results in this area on a number of problems having to do with the completion of uniform spaces. One of the theorems in this paper gives a necessary and sufficient condition for a uniform space to be complete in terms of the radical of a uniform space. The concept of the radical is based on the concept of the extent of a covering in ~X. If (X, Jl) is a uniform space, the radical of (X, Il) is defined to be nlEu IU E Ill. i.e., the intersection of all the extents in ~X of uniform coverings. This theorem states that if R is the radical of (X, Jl) then X is complete if and only if R = X. We will refer to this result as Tamano's Completeness Theorem. This section is devoted to its proof. LEMMA 6.10 Let A be a regularly open set in a space X. Let B be an open set which is not contained in A. Then there is an open C c B with CnClx(A) = 0. Proof: If Be Clx(A) then B c Intx(Clx(A)) = A since A is regularly open. But B is not contained in A so B is not contained in Clx(A). Then C = Bn[X - Clx(A)] is the desired subset of B such that CnClx(A) = 0.LEMMA 6.11 Let X be a dense subspace of a space Y. Then (1) If A is regularly open in Y then AnX is regularly open in X, (2) B is regularly open in X if and only if B = Inty(Cly(B))nX, (3) if A is regularly open in Y and B is open in Y, and if AnX contains BnX then B c A and (4) Two regularly open sets A,B in Yare identical if and only if AnX=BnX. Proof: Assume A is regularly open in Y.
AnX c
Intx(Clx (A
nX»
172
6. Paracompactifications
Inty(Cly(A nX))nX by Lemma 6.2. Now Inty(Cly(A nX))nX c Inty(Cly(A))nX = A nX since A is regularly open in Y. Therefore, A nX = Intx(Clx(B))nX so A nX is regularly open in X. This proves (1).
To show (2) suppose B = Inty(Cly(B))nX. Since Inty(Cly(B)) is regularly open in Y, by (1) above, B is regularly open in X. Conversely, suppose B is regularly open in X. Then B = Intx(Clx(B)) so by Lemma 6.2, B Inty(Cly(B))nX. Therefore, B is regularly open in X if and only if B = Inty(Cly(B))nX. To show (3) assume A is regularly open in Y and B is open in Y. Suppose B is not contained in A. By Lemma 6.l0 there is an open C c B with C rIA = 0. Since X is dense in Y, there exists apE C such that p E X. But then p E B nX but p is not in A nX which implies B nX is not contained in A nX. Therefore, B nX c A nX which implies Be A. To show (4) assume both A and B are regularly open in Yand A nX = B nX. Then by (3) above A c Band Be A so A = B. Conversely, if A = B then A nX = B nX. Hence two regularly open sets A, B in Yare identical if and only if A nX =BnXTamano worked exclusively with entourage uniformities. His original proof was somewhat different than the proof we give here but the proof given here preserves his approach. The following development is a good example of how working with covering uniformities leads to simpler statements of theorems as opposed to working with entourage uniformities. Tamano's proof was subdivided into a number of propositions. Each of the next five propositions, with the exception of Proposition 6.11, is equivalent to one of his propositions. PROPOSITION 6.6 Let X be a dense subspace of a Tychonoff space Y and let A be a basis for a uniformity ~ consisting of regularly open coverings. For each U E A, Star(VE(y),UE(Y)) c [Star(V,UW(y) so V <* U implies V€(Y) <* u£(Y) for each V E A. Proof: Since each U E A consists of regularly open sets, UE(Y) = {lnty(Cly(V)) IV E U} by Lemma 6.4. Suppose y E Star(VE(Y) , UE(Y)). Then there exists a VIE U such that y E vt(Y) and Vt(Y)nV£(y) "# 0. Therefore, there is a Z E Vl(y) nVE(Y) nX so VIC Star (V, U). Now Y E Vl(y) implies y E Inty(Cly(V d) c Inty(Cly(Star (U, U))) c [Star (V, UWCYl. Next let U, V E A with V <* U. Then Star (V, V) c V for some V E U. By the previous argument, Star (VE(Y) ,VE(Y)) c [Star (V, VW(y) and since Star (V, V) c V, we have [Star (V, VW(Y) c VE(Y). Therefore, Star (VE(Y), VE(Y)) c VE(Y) so VE(y) <* UE(y)._
Let (X, ~) be a uniform space and (X', ~') its completion. Let ~X and ~X' be the Stone-Cech compactifications of X and X' respectively. Then ~X' is also a compactification of X so by Theorem 6.1, there exists a continuous function <1>:
6.3 Tamano's Completeness Theorem ~X ~ ~X' such that <1> is onto, <1>(X) regularly open basis for~.
173
= X and
<1>(~X - X)
= ~X' - X.
Let A he a
PROPOSITION 6.7 For each U E ~, UE(~X') = U'E(~X') Proof: Let U E ~. It suffices to show that UE(~X') = U'E(~X') for each U E U. Since X is dense in X' which is in turn dense in ~X', by Lemma 6.1, U = u'nX which implies U' c UE(X'). Therefore, U'E(~X') c [UE(X')jE(j3X') = UE(~X') hy Proposition 6.1.(2). But U c U' which implies U E(j3x') c U'E(j3X'). Therefore, U E(j3x') = U'E(j3X')._ PROPOSITION 6.8 For each U E A, <1>-1 (U'E(~X') < W(j3X). Proof: Let U E A. It suffices to show that <1>-1 (U'£(j3X') c UE(~X) for each U E U. By Proposition 6.7, U'E(~X') = U£(j3x') so <1>-1 (U'E(~X') = <1>-1 (UE(~X'). Now
is open in ~X and U£(j3X) is regularly open in ~X since U E A. Therefore, by Lemma 6.11.(3) it suffices to show that <1>-1 (U£(j3X')nX c U£(j3X)nX = U. For this let y E <1>-1 (U E(j3X')nX which implies Y E X and <1>(y) E U£(j3x'). Now Y E X implies <1>(Y) = Y E U£(j3x')nX = U. We conclude that <1>-1 (U E(j3x') )nX c U. -
<1>-1 (U£(j3X')
LEMMA 6.12 R
= ndu{S(x,U£(~X)) Ix E
Xl)
= ndu{S(x,Ul(j3X) Ix E
Xl). Proof: From the definition of the radical, R = n A[u {U£(~X) I U E U I) c n A[u{S(x,W(j3X)Ix E Xl) c n A[{uS(x,Ul(j3X) Ix E Xl). Conversely, suppose Y does not belong to ndu{S(x,Ul(j3X) Ix E Xl). Then there exists a V E A such that y does not belong to S(x,V l(j3X) for each x E X which implies y does not belong to U E(j3X) for each U E U which in turn implies y does not belong to R. Hence, R = n A[u{S(x,U E(j3X) Ix E Xl) = n A[u{S(x,W(j3X) Ix E Xl).PROPOSITION 6.9 A point p E ~X is contained in the radical if and onlyif<1>(p)E X'. Proof: If pER then for each U E A, p E Sex, W(~X) for some x E X. Put \jf(U) Hence K u = S(p, U E(j3X)nX '# 0. Let V(P) be an open neighborhood of p in ~X. Then V(p)nS(p,U£(j3X) '# 0 which implies V(p)nK u '# 0. Consequently, p E Clj3x[K u l for each U E A. Since <1> is continuous, <1>(P) E Cl j3x,[K ul. We wish to show that nACl j3x,[K ul is a single point q E X' which will mean <1>(P) EX'.
=x.
For this consider the net \jf:A ~ X. For each U E A, \jf(U) E K u. Let V, A such that W* < V* < U. Then p E S(\jf(W),WE(j3X)nS(\jf(V),V Ec!>X) which implies [S(\jf(W),WW(!3X)n[S(\jf(V),VW(j3X) '# 0 which in turn implies S(\jf(W),W)nS(\jf(V),v) '# 0. Hence \jf(W) E Star (S(\jf(V),v),V) c S(\jf(V),U). W
E
6. Paracompactifications
174
So, 'l' is Cauchy in X and in X'. Therefore, 'l' converges to some q want to show that {q} = nACI~x,[K ul. For this let F u E
U
= {'l'(V) I V
E
A and
V*
E
X'. We
< U). Then F u c K u for each U
A. 'l' converges to q implies q E Clx,(F u) c Cl ~x,(F u) c Cl ~x,(K u) for each E A. Therefore, q E Cl ~x'(K u). Suppose y =F q is another point of ~X'. Then
there exists an open A c ~X' containing q such that y does not belong to Cl ~x'(A). Then AnX' is an open set in X' containing q so there exists a U' E A' with S(q,U') c AnX'. Let V' E A' with V' <* U'. 'l' converges to q implies there exists a w~
E
A' with W;) <* V such that
W' E A' with W' <* W~ which implies 'l'(W) E S(q,V') for each W' E A' with W' <* W~. Now 'l'(W) E S(p,WE(~X) so P E S('l'(W).WE(~X). For each x E K w. P E S(X,WE(~X) so there exists WI, W 2 E W with p E W1(~X)nW~(~XJ and 'l'(W) E W) uW 2. Since W1(~X)nW~(~x) =F 0, W! nW 2 =F 0 which implies there
exists a V) E V with W) uW 2 C V). Since 'l'(W) E S(q,v'). there exists a V~ E V' such that q. 'l'(W) E V~. Then V')nV~ =F 0. Consequently, there exists a U' E U' with V') uV~ c U'. Then q, x E U' which implies K w c Seq, U') c AnX' c A. Hence y does not belong to K w which implies y does not belong to = {q}.
nACl~x'[K uj· Therefore, nACl~x'[K ul
Conversely, if <jl(P) E X' then <jl(P) E U' for some U' E U', for each U' E A'. But then <jl(P) E U'E(~X') C UE(~X) by the proof of Proposition 6.8. Hence p E E u for each U E A which implies pER. We conclude that pER if and only if <jl(P) EX'.-
THEOREM 6.8 if and only if R =X.
(H.
Tamano, 1960) A uniform space (X,I1) is complete
Proof: Assume X is complete. It is evident that X cR. Let pER and for each U E A let 'l'(U), K u and F u be defined as in the proof of Proposition 6.9. Let V E A such that V* < U. By Proposition 6.6, VE(~X) <* UE(~X). Let y E K v = S(p,VE(~X)nX. Then there exists a V) E V with y, p E Vj(~X) and a V2 E V with P,'l'(V) E V~(~X). Then Vj(~X) n V~(~X) =F 0 and since VE(~X) <* UE(~X). there exists aU E U with Vj(~X)uVi(~X) c UE(~XI Therefore, Y E UE(~X)nX = U so Y E S('l'(V),U) for each V E A with V* < U. Hence. F v c K v c S('l'(V). U) for each V E A with V* < U, so 'l' is Cauchy in X. Since X is complete 'l' converges to some q E X. By an argument similar to the proof of Proposition 6.9, n{ Cl ~x[K ull U E A} = {q} and p E Cl ~x[K u 1for each U E A. Therefore, p = q E X so R = X. 0
Conversely, assume R = X and suppose (X, 11) is not complete. Then there exists a point q E X' such that q does not belong to <jl(X). Let p E pX such that <jl(P) = q. Then pER by Proposition 6.9. Since p does not belong to X we have R =F X which is a contradiction. We conclude that (X, 11) is complete. -
6.3 Tamano's Completeness Theorem
175
A Tychonoff space X is said to be topologically complete if there exists a uniformity 11 for X such that (X, 11) is complete. Clearly a Tychonoff space is topologically complete if and only if it is complete with respect to its finest uniformity. THEOREM 6.9 (H. Tamano, 1960) X is topologically complete only if for each p E ~X - X there is a partition of unity $ Cl ~x(O(<1>,,)) does not contain pfor each <1>" E $.
= {<1>,,}
if and
such that
Proof: Assume X is complete with respect to a uniformity 11, and that p E ~X X. By Tamano's Completeness Theorem, X = n{E u I U Ell\. Then, there is a U E 11 such that p E ~X - E u. Let (U,,) be a normal sequence with U 1 <* U.
Let d be the pseudo-metric on X defined (as in the proof of Theorem 2.5) with respect to {U,,}. Then the collection of spheres of radius 2-" coincides with the point-star coverings associated with U" [i.e., Sex, 2") = sex, Un) for each nJ and d:X x X ~ [OJ]. Let 't denote the topology on X induced by d. By Stone's Theorem, (X,'t) is paracompact. Now consider the open covering V = (V(x) Ix EX} of (X,'t) where Vex) = (y E X I d(x,y) < li41 and let {V d be an open locally finite refinement of V. By Theorem 6.3 there exists a partition of unity $ = {<1>,,} subordinate to (V" I such that 0(<1>,,) c V" for each <1>" E $. Since 't is coarser than the original topology, $ is also a partition of unity with respect to the original topology. We now show that p does not belong to Cl ~x(V(x» for each x E X which implies p does not belong to CI~x(O(<1>,,)) for each <1>" E $ which will establish the necessity part of the theorem. We can show this by demonstrating that Cl ~x(V(x» c E u for each x E X. For this let Z E X and for each x E X put dz(x) = d(z, x). Then dz:X ~ [0,1] is uniformly continuous with respect to the pseudo-metric uniformity on X by an argument similar to the one in (1) ~ (2) of Theorem 6.6. By Theorem 6.2, d z can be uniquely extended to a continuous function dr~X ~ [0,1]. Let r E Cl ~x(V(x». Then d~(r) .,; 1/4 so there exists an open W(r) in ~X containing r such that d~(y) < 1/2 for each y E W(r)nX. Therefore, W(r)nX c S(z, 1/2) = S(z, U 1) c U for some U E U. But then W(r) c Cll3x(U) c Eu. Hence CIBx(V(z» c Eu. This completes the necessity part of the proof. To prove the sufficiency part of the theorem, let $ = (<1>,,) be a partition of unity on X such that p does nor belong to Cl BX(O«!>A)) for each <1>" E $ where p E ~X - X. For each x E X and each positi~e integer n, put Vn(x) = ( Y E: X I .q I<1>,Jx) - <1>" (y) I < 2- n I and let Vn = IV n (x) Ix EX}. Then {V,,} is a normal sequence in X, so Vn E U for each positive integer n. Let p E ~X - X and let x E X. Then <1>,Jx) = 0 for all but finitely many (!>A's, say <1>"l ...
176
6. Paracompactifications
m} or else L7'=l I<1>", (x) - "i (y) I = L7'=l I<1>", (x) I = 1. Therefore, Y E 0(<1>,,) which implies Vn (x) C U7'=l 0(<1>,,) which in turn implies CI ~x [Vn (x) 1 c CI~X[U7'=lO(,,)], so p does not belong to CI~x[Vn(x)J. Therefore, p does not belong to R (the radical) with respect to the u uniformity. Hence R = X. By Tamano's Completeness Theorem, (X, u) is complete. Consequently, X is topologically complete. Let X be a Tychonoff space. For each f E C*(X) define If I to be sup{lf(x)lxE X}. Then I I iscalledthesupremumnormonC*(X). We can define a metric don C*(X) by d(j,g) = If - gl = sup{ If(x) - g(x) I Ix E X}. It can be shown (Exercise 1) that (C*(X), d) is a complete metric space. The following lemma appeared in Irving Glicksberg's 1959 paper Stone-Cech Compactijications of Products (Transactions of the American Mathematical Society, Volume 90, pp. 369-382). LEMMA 6.13 (I. Glicksberg, 1959) Let X and Y be Tychonoff spaces and let f E C*(X x Y). If the mapping :Y ~ C*(X), defined by (y) = fy where fix) =j(x,y) for each x E X, is continuous, then f has a continuous extension f~ to ~Xx Y. Proof: For each y E Y, let ft denote the extension of fy from X to ~X, and put ~(Y) =ft· Then ~:Y ~ C*(~X). Now C*(~X) with the metric d~ generated by the supremum norm is isometric with C*(X) with the metric d. To see this, note that there is a natural one-to-one correspondence 8:C*(X) ~ C*(~X) defined by 8(f) = f~. It is easily shown (Exercise 2) that d~(j~, g~) = d(j,g) for any pair f, g E C*(X) so 8 is an isomorphism. Consequently, the continuity of implies the continuity of ~ . Notice that for each (x,y) E X x Y that f(x,y) = fY(x). This motivates our definition of f~ as follows: for each (x,y) E ~X x Y putf~(x,y) = ft(x). Clearly, f~ is an extension of f, Since sup {ft (x) Ix EX} = sup {fy(x) Ix EX} for each y E Y, we have thatf~ is bounded. To show f~ is continuous, let (xo,Yo) E ~X x Y and let £ > O. Since fto E C*(~X), there exists a neighborhood U of x 0 in ~X with Ifto(x) - fto(xo) I < £/2 for each x E U. Since ~ is continuous, there is a neighborhood Y of Yo in Y with d~(~(Y) - ~(Yo» < £/2 for each y E V. Therefore, 1ft - fto I < £/2 for each y E Y, so Ift(x) - fto(x) I < £/2 for each x E ~X. Thus for each (x,y) E U X Y, If~(x,y) - f~(xo,Yo)1 = Ift(x) - fvo(x) I $ Ift(x) - fto (x) I + Ifto (x) - fto (x 0) I < £/2 + £/2 = £, which establishes the continuity of f~ at (xo,Jo). Since (xo,Yo) was chosen arbitrarily,f~ E C*(PX).
-
6.3 Tamano's Completeness Theorem
177
THEOREM 6.10 (H. Tamano, 1962) The following conditions on a Tychonoff space are equivalent: (1) X is topologically complete. (2) For each p E ~X - X there is a normal sequence {Vn} of open coverings of X such thatfor each Va E Va, Cl~x(V 0) does not contain p. (3) If P E ~X - X then (p) x X and M are functionally separated (by a member of C*(X x ~X)). Proof: Assume X is topologically complete and let p E ~X - X. By Theorem 6.9, there exists a partition of unity = {
But then Cl ~x[V o(x)] does not contain p for each x E X. Thus (1)
~
(2).
To show that (2) ~ (I), let {Vn} be a normal sequence of open coverings of X such that for each V 0 EVa, Cl ~x(V 0) does not contain p. Using Corollary 2.2, we can construct a pseudo-metric d on X such that W < Vo where W = {W(X)IXE X} andW(x)= lYE Xld(x,y)< I} foreachxE X. Let-rdenotethe topology of (X, d). Then (X,-r) is paracompact. Clearly W(x) is open with respect to -r and W(x) does not contain p for each x E X. Since (X,-r) is paracompact, we can use Theorem 6.3 to construct a partition of unity = {
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defined by (y) = dy for each y E X, is continuous. Therefore, by Lemma 6.13, d has a continuous extension d*(z,y) over pX x X. It is clear that d* = 1 on {p} x X and d* = 0 on L1X. Thus (1) ~ (3). To show (3) ~ (1), let P E PX - X and let {p} x X be functionally separated from L1X by f E C*(PX x X). Without loss of generality, we may assume f(L1X) = 0 and f( {p} x X) = 1. For each x E X let fx be defined by fxCz) = f(z,x) for each z E pX. Then put d(x,y) = SUpzE!3X IfAz) - fy(z) I. Then d is a pseudo-metric on X. Let 't denote the topology of (X, d), and consider the space (X,'t) which is paracompact. For each x E X, let Vex) = {y E Xld(x,y) < 1/2}. Then Vex) is open in (X,'t). Put U = {V(x) x EX}. Then there exists a partition of unity <1> = {A} that is subordinate to U. Since 1: is coarser than the original topology, U is an open covering with respect to the original topology and <1> is a partition of unity with respect to the original topology. 1
Now d(x,y) < 1/2 implies that SUpzE!3X IfAz) - f/z) 1 < 1/2 so Ifx(y) - fy(y) 1 < 1/2. Butfy(Y)=f(Y,y)=Oso Ifx(y) < 1/2. Therefore, fAz)~ 1/2foreachz 1
E Cl 13X[V(x)] since fx is continuous on pX. On the other hand,fx(P) = f(P, x) = I for each x E X. It follows that CI 13x [V(x)] does not contain P for each x E X. Since <1> is subordinate to U, CI 13x [O(A)] does not contain P for each <1>1.. E <1>. By Theorem 6.9, this implies X is topologically complete.EXERCISES 1. Let d be the metric constructed from the supremum norm on C*(X) for a Tychonoff space X. Show that (C*(X), d) is a complete metric space. 2. Show that d 13 (f13,g 13) = d(f, g) in the proof of Lemma 6.13, for any pair f, g E C*(X).
6.4 Points at Infinity and Tamano's Theorem Although Tamano's Theorem is not about uniform spaces, the proofs of Tamano's Completeness Theorem and Tam an 0 ' s characterization of topological completeness (Theorem 6.9) hold the key to its proof. Also, it is an important example of using points at infinity to prove things about the space itself. If we consider the space R of real numbers, we notice that R has a compactification R* that is constructed by adding two additional points that we denote by _00 and +00. R is sometimes denoted by (-00, +(0) while R* is often referred to as the extended real numbers and denoted [-00, +00]. The point -00 is assumed to precede all members of R and +00 is assumed to follow all members of R. In this way, the natural ordering on R is extended to R*. The topology of R* is formed by taking as a basis, all open intervals in R together with all sets in R* of the form [-00, r) and (r, +00] where [-00, r) = {y E R* 1 y < r} and (r, +00] = {y ER*ly>r}.
6.4 Points at Infinity and Tamano's Theorem
179
The points -00 and +00 are called points at infinity. In general, if (X, Jl) is a unifonn space and (X', Jl) is its completion, the set X' - X is called the set of points at infinity of X. Similarly, if P is a paracompactification of X, PX - X is referred to as the set of points at infinity of X. Points at infinity are always relative to some uniformity. Points at infinity with respect to precompact unifonnities (e.g., ~X - X) play an important role in general topology. If Jl is a unifonnity for X and H C JlX - X, then H is called a set at infinity. Compact sets at infinity will be important to us in what follows. Sets at infinity can be very large with respect to the original unifonn space. For example, the space N of positive integers with the usual topology is countable whereas ~N - N is not. We have already seen a characterization of local compactness in tenns of compact sets at infinity, namely, Proposition 6.4. Our next characterization of paracompactness in tenns of compact sets at infinity relies on a well known characterization of paracompactness by E. Michael that appeared in the Proceedings of the American Mathematical Society in 1953 (Volume 4, Number 3, pp. 831-838). LEMMA 6.14 (E. Michael, 1953) The following properties of a regular topological space are equivalent: (1) X is paracompact, (2) every open covering of X has a locally finite (not necessarily open) refinement and (3) every open covering of X has a locally finite closed refinement. Proof: (1) ~ (2) is obvious. To show (2) ~ (3) let U be an open covering of X. Since X is regular, there exists an open covering V of X such that the closures of elements of V is a refinement of U. By assumption, there exists a locally finite refinement W of V. Then the closures of elements of W is the desired locally finite closed refinement of U.
To show that (3) ~ (1), let U be an open covering ofX. Let V be a locally finite refinement of U and let W be a covering of X consisting of closed sets, each one of which intersects only finitely many members of V. Now let Z be a locally finite closed refinement of W. For each V E V, let V' = X - u{Z E Z IVnZ = 0}. Then V' is an open set containing V such that if Z E Z, then Z intersects V' if and only if Z intersects V. For each V E V pick a U v E U such that V C U v· Let U' = {V' nUv I V E V). Then U' is an open refinement of U and since each element of a locally finite covering Z intersects only finitely many elements of U', U, is locally finite. Notice the similarity between statement (2) of the following theorem and the statement of Theorem 6.9. By replacing points at infinity with compact sets at infinity in the hypothesis, the strength of the conclusion is raised from topological completeness to paracompactness.
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THEOREM 6.11 (H. Tamano. 1960) Let BX denote any compactification of X. Then the following statements are equivalent: (1) X is paracompact. (2) For each compact set C at infinity. there exists a partition of unity
To show (3) ~ (1), we use Lemma 6.13. Let {V 13} be an open covering of X and for each ~ let V~ denote the proper extension of V 13 over BX. Put C = BX - uV~. Then C is a compact set at infinity. By hypothesis there is a collection {G a.} of compact subsets of BX with X c uG a., CnG a. = 0 for each a, and for each x E X there is an open neighborhood V'(x) of x that meets only finitely many of the Go. 's. Now each G a. is covered by a finite number of the V~'s, say vy ... V~. Put H~ = Go.nV~nX. Then each H~ is contained in some VI3 = V~nX and Go.nX = uZ=IH~. Constructing the H~'s for each Go. in this manner yields a locally finite refinement {H~} of {V 13 }. It follows that X is paracompact. Let C(X) denote the collection of all real valued continuous functions on a space X and let C'(X) be a subset ofC(X). If there is anfE C'(X) such thatf(x) = 1 for each x E F and f(x) = 0 for each x E G, then F and G are said to be functionally separated by a member of C'(X). The following theorem contains Tamano's Theorem, i.e., (1) and (3) are equivalent. THEOREM 6.12 (H. Tamano, 1962) Let BX be any compactification of X. Then the following statements are equivalent: (J) X is paracompact. (2) For each compact C c ~X - X, there is a normal sequence of open coverings {V n I of X such that for each V 0
E
Va. CIBx(V o)nC = 0.
(3) X x BX is normal.
(4) IfG = X x C is closed in X x BX and GnlV( = 0, then G and IV( are functionally separated (by a member of C*(X x BX)).
6.4 Points at Infinity and Tamano's Theorem
181
Proof: The proof of the equivalence of (1) and (2) is almost identical to the proof of the equi valence of (1) and (2) in Theorem 6.10 and therefore will not be included here. It consists of exchanging the point p E ~X - X with the compact set C c ~X - X and adjusting the proof accordingly.
To show that (1) ---+ (3) we use the original proof of Dieudonne (1944). For this let U be an open covering of X x BX where X is assumed to be paracompact. For each point (x,y) E X x BX there are neighborhoods V(x,y) of x in X and W(x,y) of y in BX with (x,y) E V(x,y) X W(x,y) c U for some U E U. For a fixed x E X, the sets V(x,y) x W(x,y) such that y E BX form an open covering of the compact set {x} x BX. Therefore, there exists a finite number of points y 1 ... Yn E BX such that {V(X,Yi) x W(X,Yi) Ii = 1 ... n} covers {x} x BX. Put Vx = n;'=1 V(X,Yi). Then {Vx x W(x,y;) I i = 1 ... n} is a covering of {x} x BX and Vx is a neighborhood of x in X. Let V = {Vx Ix EX}. Then by assumption there exists a locally finite open refinement V' of V. For each V' E V' pick xv' E V'. Then XV' E V' C VXo for some Xo E X. Let Y 1 ... Ym be the finite collection of points in BX corresponding to x 0 such that {Vxo x W(XO,Yi) Ii = 1 ... m} covers {xo} x BX. Then the collection {V' x W(xv"Y;) Ii = 1 ... m} covers {xV'} x BX. Hence U' = {V' x W(xv' ,y;) IV' E V} is an open covering of X x BX that clearly refines U. Furthermore, U' is locally finite since if (a,b) E X x BX, there exists an open set A in X containing a such that A meets only finitely many members of V'. But then A x BX meets only finitely many members of U'. This completes the proof that (1) ~ (3). To prove (3) ~ (4) note that LlBX (the diagonal of BX) is closed in BX x BX so that LlBXn[X x BX] is closed in X x BX. But LlBXn[X x BX] = LlX, so LlX is closed in X x BX. Since X x BX is normal, there exists a continuous function f E C*(X x BX) with f(LlX) = 0 and f(G) = 1, so G and LlX are functionally separated. The proof that (4) ~ (1) is very similar to the proof the (3) ~ (1) of Theorem 6.10 and therefore will not be included here. It consists of exchanging the point p in ~X - X with the compact set C c ~X - X, and adjusting the proof accordingly. •
EXERCISES 1. Show that (1) and (2) of Theorem 6.12 are equivalent.
2. Show that (4) implies (1) in Theorem 6.12. 3. [A. Hohti, 1981] A uniform space (X, Jl) is uniformly paracompact if and only if for each compact K c ~X - X there is a U E Jl with Cll3x(S(x,U» c ~X K for each x E X.
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6. Paracompactifications
4. lA. Hohti, 1981] A uniform space (X, IJ.) is unifonnly paracompact if and only if (X x ~X, IJ. x ~) is C-nonnal. C-nonnality is defined as follows: (X, IJ.) is C-normal if for each pair A, B of disjoint closed sets, there exists a U E IJ. such that given any x E X, there is a U x E IJ. with Star(A,Ux)nS(x,U)nStar(B,U x ) =
0.
6.5 Paracompactifications
If there is a paracompactification n X of X that plays a role analogous to the role of the Stone-Cech compactification and the Hewitt realcompactification in the theories of compactification and realcompactification respectively, what properties should 11: have in order that it be considered to play an analogous role? First, if X is a paracompact space then we should expect that 1I:X = X and second. 1I:X should be realizable as the completion of X with respect to some uniformity 11: such that 11: is the finest uniformity to have a paracompact completion. Every paracompactification of X is realizable as the completion of X with respect to some uniformity as shown in Section 6.1. We will show that in general, such a paracompactification does not exist. and that a necessary and sufficient condition for 11: to exist is for the finest uniformity u to be preparacompact, in which case u = 11:. Consequently, if X is paracompact then 1I:X exists and 1I:X = X. In the case 11: exists, we call 1I:X the Tamano-Morita paracompactification of X. We begin our discussion by proving a lemma about the relationship of any Tychonoff space Y that contains X as a dense subspace and the subspaces of ~X that contain X. Let X and Y be Tychonoff spaces. A function f:X - j Y is said to be a perfect mapping if f is closed, continuous and the inverse image of each point of Y is compact in X. Let i:X - j Y be an imbedding of X into a dense subspace of Y. Let j: Y -+ PY be the imbedding of Y into its Stone-Cech compactification and let k: PX -+ PY be the unique continuous extension of j © j such that k(~X - X) = ~Y - X. Finally, let Z = k- 1 (Y). LEMMA 6.15 (J) The restriction k z = k I Z is a closed mapping, (2) Z is the largest subspace of ~X to which j © i has a continuous extension to Y, and (3) Z is the only subspace of~X containing Xfor which k z is a peifect mapping.
Proof: Since ~X and ~Y are both compact, the mapping k is closed and the inverse image of each compact set in ~Y is compact in ~X. Then kz has the same properties because it is the restriction of k to the total inverse image of Y. By definition of Z, Z is the largest subset of ~X to which j © i has a continuous extension to Y. Furthermore, we claim that Z is the only subspace of ~X for which k IZ is a closed mapping such that the inverse image of each point is compact. To show this suppose W is another subspace of ~X containing X having the property that k IW is a closed mapping and the inverse image of each
6.5 Paracompactifications
183
point is compact. Put S = WnZ. Then XeS c Z and since Z 7' W, S is a proper dense subspace of Y. Moreover, k~l(p)nS is compact for each p E Y. Choose Z E Z ~ S and let q = k (z). Then z has an open neighborhood V such that Clz(V)n[k~l (q)nSj = 0. Thus q does not belong to k(Clz(V)nS). But Clz(Clz(V)nS) = Clz(V) and q E k(Clz(V». Hence q E k(Clz(Clz(V)nS» c Cly(k(Clz(V)nS». Therefore, k(Clz(V)nS) is not closed in Y. Hence kl S takes the closed set Clz(V)nS into a set that is not closed in Y. Therefore, k Is is not a closed mapping which is a contradiction. Consequently Z is the only subspace of BX containing X for which the extension k of j © i is a closed mapping such that the inverse image of each point is compact. The following development of the Tamano~Morita paracompactification appeared in the author's paper titled Paracompactifications. Preparacompact~ ness and Some Problems of K. Morita and H. Tamano that appeared in Questions and Answers in General Topology. Volume 10, pp. 191-204.
THEOREM 6.13 Each completion J.!X of a uniform space (X. J..I.) where J..I. isfmer than B is homeomorphic to a subspace ofBX containing X.
Proof Let i:X ---+ flX be the imbedding of X into its completion and }:flX---+
PflX the imbedding of flX into its Stone-Cech compactification. By the above lemma, there exists a unique continuous extension k of} © i such that kCPX - X) = BJ..I.X - X and such that if Z = k~l (J..I.X) then (1) the restriction kz = k IZ is a closed mapping and (2) the inverse image of each point under kz is compact. Let J..I.' denote the uniformity of J..I.X and let J..I.+ be the coarsest uniformity on Z that makes kz uniformly continuous. Then a basis for J..I.+ is the collection i kZ1 (U') I u' E J..I.' I. Clearly X is dense in Z. To see that (X, J..I.) is a uniform subspace of (Z, J..I.+), let U" E J..I.+ such that U'" = k Z1(U') for some U' E J..I.'. Since kz keeps X pointwise fixed, U+ nX = U' nX = U for some U E J..I.. Conversely, if U E J..I. then there exists a U' E J..I.' such that U = U'nX. But then U = kZI (U')nX and kZI (U') E J..I.+. Consequently (X, J..I.) is a dense uniform subspace of (Z, J..I.+). Clearly J..I.+ generates the same topology on Z as the one induced on Z by BX since J..I.+ is finer than B. Finally we claim that (Z, J..I.... ) is complete and therefore (Z, J..I.+) is the completion of (X, J..I.). Since completions are unique, we can identify J..I.X with Z so that X c J..I.X c BX. To see (Z, J..I.... ) is complete. let \jf:D ~ Z be Cauchy with respect to J..I.+. Since kz is uniformly continuous, kz © \jf is Cauchy in J..I.X and hence converges to some q E J..I.X. Suppose \jf does not cluster to any p E k Z1 (q). Since k Z1 (q) is compact, there exists an open set 0 containing k Z1 (q) such that \jf is eventually in Z - O. Then kz(Z - 0) is closed in J..I.X and does not contain q. Then J..I.X - kz(Z - 0) is an open set containing q such that kz © \jf is eventually in its complement. Therefore'll does not converge to q which is a contradiction. Hence'll must cluster to some p E kZ1 (q). Since'll is Cauchy, 'II converges to p. Therefore, (Z, J..I.+) is complete. But then (Z, J..I.+) is the completion of (X, J..l).-
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PROPOSITION 6.10. The Stone-Cech compactification p~ of a J..lX c ~X of X with respect to some uniformity /-l of X is ~X (i.e., ~J..lX
completion = ~X).
Proof: Let i:/-lX ~ ~X be the identity imbedding of /-lX into ~X and let j be the identity mapping on ~X. By Lemma 6.14, there exists a unique continuous extension k:~/-lX ~ ~X such that k(~/-lX - /-lX) = ~X - /-lX and if Z - k- 1 (~X) = ~/-lX then k IZ = k is a closed mapping such that the inverse image of each point is compact. We claim that by an argument similar to the one in the proof of Theorem 6.13, we can see that ~X is homeomorphic to ~/-lX.
For this let Wdenote the uniformity of ~X and put W= ~~ (the coarsest uniformity on ~/-lX that makes k uniformly continuous). Then (X, ~) is a dense uniform subspace of (~/-lx' W) by an argument similar to the one in the proof of Theorem 6.13 that showed (X, /-l) to be a dense uniform subspace of (Z, /-l). Finally, we claim (~/-lX, W) is complete by an argument similar to the one in the proof of Theorem 6.13 that showed (Z, /-l') to be complete. Since completions are unique, ~/-lX is homeomorphic to ~X (i.e., we can identify ~/-lX with ~X so that ~/-lX = ~X.) -
If X is a subspace of Y such that each f E C(X) can be extended to a continuous function over Y, then X is said to be C-imbedded in Y. If each g E C*(X) can be extended to a continuous function over Y, then X is C*-imbedded in Y. Then every closed subspace of a normal space is C*-imbedded (by Theorem 0.4). The following theorem has many applications. Notice that Proposition 6.10 is a special case of the equivalence of (5) and (6) in this theorem. THEOREM 6.14 Let X be dense in the Tychonoff space Y. Then the following statements are equivalent: (1) Every continuousf:X ~ K, where K is a compact Hausdorff space, has a continuous extension fY'Y ~ K. (2) X is C*-imbedded in Y (3) Any two disjoint zero sets in X have disjoint closures in Y. (4) IfZ, z' E Z(X) then Cly(ZnZ') = Cly(Z)nCly(Z'). (5) ~X = ~Y. (6) X eYe ~x.
Proof: (1) ~ (2) If f E C*(X) then f is a continuous mapping from X into the compact subset C I R (f(X». Hence (2) is a special case of (1), so (1) implies (2).
(2) ~ (3) Let Z, Z' E Z(X) with ZnZ' = 0. Then there exists f, f' E C*(X) such that Z = Z(j) and Z' = Z(f'). Let g = maxI IfI,11 and g' = 1 maxI 1f'1 ,11. Put h' = minlg,g'l and let h = maxlh',ll. Then 0 ~ h ~ 1, h(Z) = o and h(Z') = 1. Let h Y be the continuous extension of h to Y. Let U = {Y E Y Ih(y) < 1/21 and V = lyE Y Ih(y) > 1/21. Then U and V are disjoint open sets
6.5 Paracompactifications
185
in Y such that Z c U and Z' c V. Consequently, Z and Z' have disjoint closures in Y. (3) ~ (4) Clearly Cly(ZnZ') c Cly(Z)nCly(Z,). To show the reverse inclusion, let p E Cly(Z)nCly(Z,). Then there exists a zero set neighborhood V of p in Y. P E Cly(V nZ) and p E Cly(V nZ'). Thus (3) implies (V nZ)n(V nZ') i= 0 which in tum implies Vn(ZnZ') i= 0. Therefore, p E Cly(ZnZ'). Consequently, Cly(Z)nCly(Z,) c Cly(ZnZ').
(4) ~ (5) We show ~X = ~Y by showing that (X,~) is a uniform subspace of (Y, ~) and appealing to the uniqueness of the completion (Theorem 4.10). To do this, it suffices to show that every finite normal covering U of X can be extended to a finite normal covering U' of Y such that U = U' nX. If U is a finite normal covering of X, then it can be shown (Exercise 4) that there exists a normal sequence {Un) of finite open coverings of X such that U 1 < U, and for each positive integer nand U E Un' X - U is a zero set of X. Next, notice that we can extend statement (4) of the theorem to any finite collection Z 1 •.. Zn so that Cly(n7=1 Zi) = n7=1 Cly(Zi)' Now UE(Y) is an extension of U into Y for each positive integer n, and U~(Y) is an extension of Un into Y. Furthermore, by the extended version of (4),
we have for each positive integer n, n{Y - VE(y) IV E Un) = n{ Cly(X - V) IV E Un) = Cly(n{ X - V IV E Un)) = Cly(0) = 0. Consequently. U~(Y) is a covering of Y for each positive integer n, so WrY) is a covering of Y by Proposition 6.6. Also, by Proposition 6.6, U n+ 1 <* Un for each positive integer n so U~(r? <* U~(Y) for each n. Hence UE(y) is a finite normal covering of Y and U = UE(y) nX. (5)
~
(6) is obvious.
(6) ~ (I) By Theorem 6.2, a continuous f:X ~ K, where K is a compact Hausdorff space, has a continuous extensionfj3:~X ~ K. But thenfY = fj31 Y is a continuous extension to Y. THEOREM 6.15 The completion of a Tychonoff space X with respect to its finest uniformity u is uX = n {J.!X IJ.! is a uniformity for X and J.!X is a subspace of~X). Proof: To see that nJ.!X is topologically complete. observe that for each point p E ~X - nJ.!X, P must belong to ~X - AX for some uniformity A of X. Since AX is complete, AX is topologically complete. By Theorem 6.9, for each q E ~AX AX, there exists a partition of unity cI> = {<1>d on AX such that C l j3AX (0(<1>,"» does not contain q for each <1>)" E cI>. By Proposition 6.10, ~AX = ~X so CL.j3,Y(O(<1>:.J) does not contain p for each <1>)" E cI>. For each <1>)" E cI> put '1')" = <1>)" InJ.!X.'Fhen 'I' = {'I'),,) is a partition of unity on nJ.!X such that Cl j3x(O('I',..» does not contain·~ p for each '1'", E '1'. Since this holds for each p E ~X - nJ.!X. we have that nJ.!X is topologically complete.
186
6. Paracompactifications
Let v be the finest unifonnity on nl..lX and let v' be the unifonnity induced on X by v. Since v is the finest uniformity on n~..IX, v is finer than the unifonnity u' induced on n~X by u. But then v' is finer than u which implies that v' = u. Since n~X is topologically complete, n~X is the completion of X with respect to v'. Since completions are unique, n~X = uX. -
LEMMA 6.16 paracompact.
A perfect preimage of a paracompact space is
Proof: Let {Va} be an open covering of X. For each y E Y, let {Va, Ii = 1 ... n(y)} be a finite subcovering for the compact set 1 (y). Then F(y) = u7=C.jl Va, is closed in X so f(F(y» is closed in Y. Put V(y) = Y - f(F(y». Then V(y) is an open neighborhood of y and rl(V(y» c u7=Q)V a ,. Next let {W(y)} be a precise locally finite open refinement of {V(y) lYE Y}. For each i = 1 ... n(y), put W(y,i) = r 1 (W(y»nV a,' Then the collection {W(y,i) lYE Y and i = 1 ... n(y)} is an open covering of X that refines {Va}. Moreover, {W(y,i)} is locally finite since if p E X, there exists a neighborhood V of f(P) that meets at most finitely many of the W(y) and then the neighborhood 1 (V) of p meets at most finitely many of the W(y,i). -
r
r
John Mack sketched an outline of the proof of the following theorem for the author late one night at the 1991 Northeast Topology Conference after a lengthy discussion of paracompact subspaces of BX containing X. It plays a major role in the author's solution to Tamano's Paracompactification Problem.
THEOREM 6.16 (1. Mack, 1991) Let u be the finest uniformity for the space X. Then for each p E BX - uX there is a paracompact Y c BX - {p} that contains uX. Proof: By Theorem 6.13, X c uX c BX so by Theorem 6.14, BuX = BX. Since uX is topologically complete and BuX = BX, by Theorem 6.10, if p E BX - uX, there exists an F E C*(uX x BX) such that F(x, x) = 0 and F(x, p) = 1 for each x E uX. Putf= max{F,l} and for each pair x,Y E uX put 8(x,y)
= sup { If(x, q) - f(Y, q) II q E
BX}.
Then max{f(x, y),f(y, x)} :<::; 8(x,y) :<::; 1. It is easily shown that 8 is a pseudometric on uX. For each pair x,y E uX define x - y if and only if 8(x,y) = 0 and set [x] = lYE uxI8(x,y)=0}. LetM= {[x]lxE uX} and put d([x],[y]) = S(x,y). Then (M, d) is a metric space and the quotient mapping ':uX ~ M defined by '(x) = [xl for each x E uX is continuous. Let (M*, d*) be the completion of (M,d).
Now put = j © i © <1>' where i:M -+ M* is the imbedding of M into its completion and j:M* -+ I3M* is the imbedding of M* into its Stone-tech
187
6.S Paracompactifications
compactification. Then :uX -7 ~M*. Let ~ be the extension of over ~uX = ~X. Since ~X and ~M* are both compact, ~ is closed and the inverse image of each compact set in ~M* is compact in ~X. Put Y = (~tl (M*) and let ~ = ~ I Y. Then ~:Y -7 M* and since Y is the total inverse image of M*, ~ is also closed and the inverse image of each compact set in M* is compact in Y. Therefore, ~ is a perfect mapping. By Lemma 6.15, Y is paracompact. Clearly, uX c Y. By a proof similar to the proof of Lemma 6.12, we can show that d*:M* x [0,1] has a continuous extension d~:M* x ~M* -7 [OJ]. Set g = d~ © (~ x ~). Then g E C*(Y X ~X) and g(Y x ~X) c [0,1]. Let (x,y) E X X X. Then g(x,y) = d~(~(x),~(Y» = d~(x,y) = d*(x,y) = S(x,y) ~f(x,y). Therefore, M*
-7
fl X x X
~ g I X x X which implies
fl X x ~X
~ g IX x ~X.
Hence 1 = f(x, p) ~ g(x, p) for each x E X. Therefore, 1 ~ g(x, p) for each y E Y. Since g(x, x) = Sex, x) = 0 for each x E X, g(y,y) = 0 for each y E Y. Consequently, p does not belong to Y. Hence Y is a paracompact subset of ~X {p) containing uX. -
THEOREM 6.17. lfrr is the finest uniformity for a space X such that the completion rrX is paracompact, then rr = u (the universal uniformity for X). Proof: Put PX = Il{Y c ~xlx c Y and Y is paracompact). Clearly PX *- 0 and since ~X is paracompact, rr is finer than ~. Therefore, rrX c ~X which implies PX c rrX. Now, for each paracompact space Y such that X eYe ~X, it is easily shown that Y is the completion of X with respect to some uniformity 'A that is finer than ~. Then by Theorem 6.15, uX c PX. Therefore, PX = ux.
There are two uniformities on X that bound rr, namely, u (the finest) and sup{'AI'AX c ~X and 'AX is paracompact), which we denote by sup'A. Clearly sup'A c rr c u. Now rrX is paracompact and PX c rrX. We claim that the completion sup'AX of X with respect to sup'A is PX. Since X c PX c 'AX for each uniformity 'A such that 'AX is a paracompact subset of ~X, 'A' (the uniformity of the completion 'AX) induces a uniformity 'A+ on PX and the uniformity 'A on X. Consequently, sup'A+ = sup{'A+) is a uniformity on PX that induces the uniformity sup'A on X. Thus (X, sup'A) is a dense uniform subspace of (PX, sup'A+). To see that (PX, sup'A+) is complete, let \jI be a Cauchy net in PX with respect to sup'A+. Then \jI is Cauchy in PX with respect to each 'A+. But then \jI is Cauchy in 'AX which implies \jI converges to some P A. E 'AX c ~X for each 'A. Since ~X is Hausdorff, p A. = P ~ for each pair of uniformities 'A, ~ such that 'AX and ~X are paracompact subsets of ~X. Therefore, there exists apE ~X such that \jI converges to p and p E PX since p = p A. E AX for each A. Hence (PX, SUPA+) is complete, and since completions are unique, SUpAX = PX.
188
6. Paracompactifications
Finally, we show that rrX = uX. Since PX c rrX, rr induces a unifonnity rr+ on PX that is finer than sup/...+ since rr is finer than sup/.... Then (X, rr) is a dense uniform subspace of (PX, rr+). Now (PX, rr+) is complete since (PX, sup/...+) is complete and rr+ c sup/...+, and since completions are unique, rrX = PX = uX. But then uX is paracompact, so by hypothesis, rr = u. Analogously to the definition of topologically complete, a space will be called topologically preparacompact if it is preparacompact with respect to its finest unifonnity.
THEOREM 6.18 (N. Howes, 1992) A necessary and sufficient condition for the existence of the Tamano-Morita paracompactification is topological preparacompactness. Proof: Suppose X is topologically preparacompact and let u be the universal unifonnity on X. Then uX is paracompact which implies that rr exists and equals u. Conversely, if rr exists, then by Theorem 6.17, rr = u which implies that uX is paracompact which in turn implies that u' is cofinally complete. Therefore, u is preparacompact which implies X is topologically preparacompact. We now show that some of the results in Chapter 4 can be generalized by replacing cofinally Cauchy nets with almost Cauchy nets. Part (I) of the following theorem is due to J. Isbell and appeared in his paper Supercomplete spaces referenced in Chapter 5.
THEOREM 6.19 (1. Isbell, 1962, N. Howes, 1992) Let X be a Tychonoff space and let u be the finest uniformity on X. Then (1) X is paracompact if and only if (X, u) is supercomplete, (2) X is LindelOf if and only if (X, e) is supercomplete and (3) X is compact if and only if (X, P) is supercomplete. Proof: (1) is essentially the equivalence of (1) and (2) in Theorem 5.8. To prove (2), we need only show that if (X, e) is supercomplete then X is Linde16f since by Theorem 4.4 and Corollary 5.1, a Lindelof space is supercomp1ete with respect to the e unifonnity. By Lemma 4.1 it will suffice to show that each (I)-directed (countably directed) net is almost Cauchy with respect to e. For this let ",:D -7 X be an (I)-directed net. Let U E e. Then there exists a countable nonnal covering {Vii that refines U. For each i put Cj = {d E D I",(d) E Vii and let E = u{ Cj ICj is not cofinal in D}. If E = 0 pick any d E D. Then {Cj } is a collection of cofinal subsets of D such that each d' ;::>: d is in some C j and for each i, ",(C j ) c V j c U for some U E U. If E "# 0, then for each i such that Cj is not cofinal in D, there exists a d j E D with R(dj)nC j = 0 where R(dj) = {d E Dldj ~ d}. Since", is (I)-directed, there exists a d' E D with d j ~ d' for each i such that Cj is not cofinal in D.
6.5 Paracompactifications
189
Hence R(d')ilC i = 0 for each i such that Ci is not cofinal in D. Therefore, R(d')ilE = 0. For each j put D j = CjilR(d'). Then {D j } is a collection of cofinal subsets of D such that each d+ ~ d' is in some D; and for each j, \jf(D j ) C Vj C U for some U E U. Since U was chosen arhitrarily, \jf is almost Cauchy. To prove (3), we need only show that if (X, ~) is supercomplcte then X is compact since by Theorem 4.4 and Corollary 5.1, a compact space is supercomplete with respect to~. For this, it will suffice to show that each net in X is almost Cauchy with respect to~. Let \jf:D ~ X be a net in X. Let U E ~. Then there exists a finite normal covering {V I ... V n} that refines U. For each i = 1 ... n put Ci = (d E D I \jf(d) E Vi}' Then an argument similar to the above can be used to show that there exists a d E D such that the collection {C i I Ci is cofinal in D) is a collection of cofinal subsets of D such that each d' ~ d is in some C i and for each i, \jf(C;) C Vi C U for some U E U. Therefore, each net in X is almost Cauchy with respect to ~. If instead of Theorem 4.4, we were to use Theorem 6.18 to motivate our definition of preparacompactness, then we would define preparacompactness to be the property that each almost Cauchy net has a Cauchy suhnet. To distinguish between these two notions we define a uniform space to he almost paracompact if each almost Cauchy net has a Cauchy subnet. Then we can prove: THEOREM 6.20 (N. Howes. 1992) Let (X. fJ.) be a Uniform space and let v be the uniformity on X derived from fJ.. Then (1) (X. fJ.) has a paracompact completion if and only if (X.V) is almost paracompact. (2) (X. fJ.) has a LindeLOf completion if and only if (X.V) is countably bounded and almost paracompact. Proof: To prove (1) let (X', fJ.') be the completion of (X, fJ.) and let u' he the finest uniformity for X'. Assume (X,v) is almost paracompact and that \jf:D ~
X' is an almost Cauchy net with respect to u'. Let E = D X u' and define < on E by (d,U') < (e,V') if d < e and V' <* U'. For each (d, U') E E put Sed, U') = a for some a E X such that a and \jf(d) both belong to some U' E U'. Then the correspondence (d, U') ~ Sed, U') defines a net S:E ~ X. Let U' E u' and pick V' E u' such that V' <* U'. Since \jf IS almost Cauchy, there exists a d E D and a collection {C a} of cofinal subsets of D such that each d' ~ d belongs to some C a and for each index a, \jf(C a) C V~ for some V~ E V'. For each index a put Ea
=
{(e,W')leE CaandW'<*V').
Then Ea is cofinal in E for each index a. Let (e,W') E Ea. Then S(e,W') = y E X such that y and \jf(e) both belong to some W' E W'. Since (e,W') E E a , e E
190
6. Paracompactifications
C a which puts \jI(e) in V~. Consequently,
y E Star(V~,W') c Star(V~,V') c Va for some V~ E U'. Therefore, 8(£ a) C Va for each index a.. To show 8 is almost Cauchy it remains to show there exists an e E E such that each e' ::::> e belongs to some £ a' For this choose W' E u' such that W' <* V' and put e = (d,W'). Now dEC j3 for some index Pwhich implies e E £ /3' If e' > e then e' = (d', Z') for some d' > d and Z' <* W'. Then d' E C y for some index yand Z' <* V' so e' E £ y' Hence 8 is almost Cauchy with respect to u'. Since 8 lies in X, 8 is almost Cauchy with respect to v and therefore has a Cauchy subnet
6.5 Paracompactifications
191
PROPOSITION 6.12. A uniform space is countably bounded if and only if each ill-directed net is almost Cauchy By a supercompletion of a uniform space (X, Il), we mean a supercomplete uniform space (y, A) such that X is uniformly homeomorphic to a dense subspace of Y. It is natural to inquire about the existence and uniqueness of a supercompletion for a given uniform space. In fact, A. Hohti, posed this question in a paper titled Uniform hyperspaces (Proc. 1983 Hungarian Topology Colloquium, pp. 333-334). The following theorem provides an answer.
THEOREM 6.21. (N. Howes, 1992) A necessary and sufficient condition for a uniform space to have a supercompletiofl is that it he almost paracompact. In this case, the supercompletion is unique and identical with the ordinary completion. The proof is left as an exercise (Exercise 3).
EXERCISES 1. Prove Proposition 6.11. 2. Prove Proposition 6.12. 3. Prove Theorem 6.21. [Hint: Adapt the proof of Theorem 4.11 using Corollary 6.1] 4. Show that if U is a finite normal covering of a Tychonoff space X that there exists a normal sequence {Un} of finite open coverings of X such that U 1 < U and for each positive integer nand U E Un' X - U is a zero set of X. RELATIVEL Y FINE SPACES In 1993, B. Burdick gave an example of a uniform space that is cofinally complete with respect to transfinite sequences but not cofinally complete (uniformly paracompact) thereby showing the existence of uniform properties that cannot be characterized by transfinite sequences in the same way they are characterized by nets. The class of relatively fine uniform spaces is a class of uniforms spaces in which the use of transfinite sequences is equivalent to the use of nets for studying several uniform properties like uniform paracompactness. Let K be an infinite cardinal. A uniformity f.l is said to be K-bounded if each uniform covering has a subcovering of cardinality less than K. For a uniformizable space X and for each infinite cardinal K there exists a finest
192
6. Paracompactifications
K-bounded uniformity. It is the supremum of all K-bounded uniformities and is denoted by UK' The uniformities UK where K is an infinite regular cardinal are called the relatively fine uniformities. Note that u, e and ~ are all relatively fine uniformities. 5. [Howes, 1994] For a relatively fine space (X, equivalent:
UK)'
the following are
(1) (X, UK) is uniformly paracompact. (2) (X, UK) is cofinally complete. (3) (X, Uk) is cofinally complete with respect to transfinite sequences. (4) X is paracompact and each transfinite sequence with no subsequence of cardinality less than K clusters. (5) X is paracompact and each K-directed net clusters. (6) Each directed open covering is uniform. 6. [Howes, 1994] Let (X, /J.) be relatively fine where /J. #-~. If there do not exist measurable cardinals then: (1) The following are equivalent: (a) (X, /J.) has a paracompact completion. (b) X is topologically preparacompact. (c) X is topologically almost paracompact. (2) The following are equivalent: (a) (X, /J.) has a LindelOf completion. (b) X is topologically preparacompact and U = e. (c) X is topologically almost paracompact and U = e. 7. [Burdick, Howes, 1994] For a locally fine space (X, /J.) the follwing are equivalent: (1) (X, /J.) is cofinally complete. (2) (X, l-l) is cofinally complete with respect to transfinite sequences. RESEARCH PROBLEM 8. What other uniform properties can be characterized in locally fine spaces by transfinite sequences and are these characterizations analogous to their net characterizations? 6.6 The Spectrum of ~X We denote the weak completion u(X) of X with respect to u, developed in Section 5.5, by uX since u(X) is homeomorphic to uX. To see this, recall that
6.6 The Spectrum of PX
193
(X, u) is unifonnly homeomorphic to a dense subspace of u(X). Hence the unifonnity of u(X) is the finest unifonnity for u(X). Then by Theorem 5.5, u(X) is complete. Since completions are unique, we have that u(X) is unifonnly homeomorphic with uX. In this section {", IA E A I will denote the collection of all nonnal sequences of open coverings of X. Let {'1\ Iy E r} be the collection of all nonnal sequences of open coverings of ~X such that 'I'y begins with a finite covering. If Y E r then 'I'y = {We I for some nonnal sequence of open coverings such that W~ is finite. Then for each n, W n = {W j3 nX IW j3 E We} is an open covering of X and
Proof: The proofs of (1) and (2) are trivial. To see (3) let x E G. Then there exists a positive integer n(x) such that S(x,Wn(x) c G. Put K = u{S(x,We(X) Ix E G}. Clearly KnX = G. Let H be the interior of K in [~Xlr. Clearly G c H c K so HnX = G. This proves (3).-
For any continuous f:X ~ Y where X and Y are both Tychonoff spaces, there exists a continuous extension fj3:~X ~ ~Y. By Lemma 6.16, X/y is a unifonn subspace of ~X/'I'y. Let 11y:X/y ~ ~X/'I'y denote the inclusion mapping. Then we have the following diagram:
II
Clearly ~X/'I'y is compact since it is the quotient of a compact space, so ~(~X/'I'y) = ~X/'I'f Now 9y restricted to X is 11y © <1>y so by the above diagram we see that 9y =11y © <1>~.
194
6. Paracompactifications
Let \fI",\fIfl be two nonnal sequences of open members of W (the unifonnity of ~X) such that A < f.!. Then, just as in Section 5.5, there exists a bonding map (<j>~)~:~X/\f'fl ~ ~X/\f'". Clearly (<j>~)~ restricted to X/<1>fl is just <j>~. Then, just as in Section 5.5, (~X/\f'd<j>~)~} is an inverse limit system of unifonn spaces, and the inverse limit of this system, which we denote by W(~X) is the weak completion of ~X with respect to W. Since ~X is complete, by Theorem 5.4, W(~X) = ~X, so ~X is the inverse limit of {~X/\f'd<j>~)~}. If \fI" is a nonnal sequence of open members of Wthen there exists a \fI fl such that f.! E r with A < f.!. To see this let \fI" = {W~} and let Y E r. Let \fly = {V~}. Then V~ is finite and {W~nV~} is a normal sequence. Put U~ = V~ and for each n > I put U~ = W~nV~. Then let f.! be the element of r such that \fIfl = {U~}. Clearly, A < f.!. Therefore, {\fly lYE r} is cofinal in the collection of all nonnal sequences of open members of W. Consequently, by Exercise 5 of Section 5.5, ~X is the inverse limit space of the inverse limit system {~X/\f'y, (<j>~)~} where YE r. Now since lly:X/<1>y ~ ~X/\f'y is the inclusion mapping, ll~ must be one-toone, so ~(X/<1>y) can be identified with the subset ll~[~(X/<1>y)] of ~X/\f'"y" But then ~X/\f'y is a compactification of X/<1>y containing a copy of ~(X/<1>y) so ~X/\f'r can be identified with ~(X/<1>y). Moreover, for each pair A, f.! E r with A < f.!, <j>~:X/<1>fl ~ X/<1>" has a unifonnly continuous extension ~(<j>~):~(X/<1>fl) ~ ~(X/<1>,,). Since ~(X/<1>fl) and ~(X/<1>,,) can be identified with ~X/\f'fl and ~X/\f'" respectively, we can consider ~(<j>~) as a mapping from ~X/\f'fl into ~X/\f'". Then ~(<j>~) restricted to X/<1>fl is the same as (<j>~)~ restricted to X/<1>w Therefore, ~(<j>~) = (<j>~)~. Consequently, we have the following: THEOREM 6.22 (K. Morita, 1970) {~(X/<1>,,), (<j>~)~} is an inverse limit system whose inverse limit can be identified with ~x. Proof: The proof of this theorem has essentially been proved in the preceding discussion. It remains to observe that since r is cofinal in A, the inverse system in the hypothesis of the theorem has the same inverse limit as the system where A ranges over r, namely, ~X (see Exercise 5, Section 5.5). Consequently, ~X has the spectrum W(X/<1>,,), (<j>")~}. Recall from Section 5.5 that uX = u(X) has the spectrum {X/<1>", <j>t}. Also recall the unifonn imbedding <j>:X ~ uX defined by <j>(x) =( <j>dx)} E uX. It's extension <j>~:~X ~ ~uX = ~X can be seen to be the mapping defined by <j>~(x) = (<j>~(x)} E
~X.
THEOREM 6.23 (K. Morita, 1970) For any uniformizable space X, uX c ~X and uX = n {(<j>~rl (X/<1>,,) IA E A} = n {(f~rl (T) I f:X ~ T is continuous with T metrizable}. Proof: That uX c
pX follows from either Theorem 6.13 or Exercise 1.
To prove
6.6 The Spectrum of PX
195
the first equality, suppose x E uX. Then x E fhX/A such that for each pair A, Il E A with A < Il, ~(xh!) = x A. Hence x E (~rl(X/A) for each A E A. Therefore, uX c n{(~rl(x/A)IA E A}. Conversely, suppose x E n{(~rl(X/A) IA E A}. Then ~(x) E X/A for each A E A. Therefore,
1
To prove the second equality, let f be a continuous mapping of X into a metric space T. For each positive integer n, let Un = {Set, 2- n) t E T}. Then the sequence {Un} forms a basis for the metric uniformity of T. For each n put Vn = 1 (Un)' Then {V n} is a normal sequence of open coverings of X, so {V n} =
r
To see that g is continuous let t E T and let U be an open neighborhood of t in T. Then there exists a positive integer n such that Set, 2- n ) c U in T which implies Set, Un) C U which in tum implies Star(f-l (t),V n) c F- 1 (U). Pick x E 1 (t). Then S(x,V n) is open in X A' Put V = S(x,v n). Then fey) c U, so (g © ~ © i A)(V) c U. Since i A(V) = V, fey) = (g © ~)(V). For any open W c X Awe have:
r
W
=
(XE XAIS(x,V n) c W for some positive integer n}.
From this it is easily seen that (~rl(A(W» = W so that ~ is also an open mapping. Hence Z = ~(V) is open in X/A and g(Z) c (g © ~)(V) =fey) c U. Moreover,
«g
1
LEMMA 6.18 fiX eYe uX then uY = uX.
196
6. Paracompactifications
Proof: If g: Y -7 T is continuous where T is a metric space, then f = g IX: -7 T is continuous. Thenr:uX -7 uT, gU:uY ~ uTand (ix)u:uX -7 uY, where i x :X-7 Y is the inclusion (identity) mapping. Since (ixt = iuX , uX c uY. Since gUlux:ux -7 uT,r = gUlux. Butr =f~lux and gU = g~luY sof~lux = (g~ I uY) IuX = g~ I uX which impliesf~ = g~. Conversely, for any continuousf:X -7 T where T is a metric space,r:uX -7 uT. Since T is metric, T is weakly complete with respect to the metric uniformity which implies T is weakly complete with respect to u so uT = T by Theorem 5.4. Therefore, g = IY is a continuous mapping from Y into T such that g IX =f.
r
Then by Theorem 6.22, uY = ill (g~rl (T) Ig: Y -7 T where g is continuous and T is metrizable) c ill (f~rl (T) If:X -7 T where f is continuous and T is metrizable) = uX. Therefore, uY c uX. Similarly, uX c uY so uY = uX.-
THEOREM 6.24 (K. Morita. 1970) uX is characterized as a space Y with the following properties: (1) Y is topologically complete and contains X as a dense subspace. (2) Any continuous f:X ~ T where T is a metric space can be extended to a continuous mapping from Y into T. Proof: uX is the inverse limit of the inverse system of metric spaces I X/cD", ~} and hence by Theorem 5.5 is topologically complete. Thus uX satisfies (1). To show uX satisfies (2), letf:X -7 T where T is a metric space. Thenr:uX -7 uT. But since T is a metric space, T is weakly complete, so by Theorem 5.4, uT = T. Therefore, is a continuous mapping from uX into T.
r
Conversely, let Y be a space satisfying (1) and (2). Let g E C*(X). Then g(X) is a bounded subset of R and hence metrizable, so g:X -7 g(X) is a mapping of X into a metric space and by (2) g can be extended to Y. But then X is C*-imbedded in Y so by Theorem 6.14, BX = BY and X eYe BX. Again by (2), for any continuous mapping f from X into a metric space T, there is a continuous extension g:Y -7 T. Since BX = BY we have f~ = g~ where f~ and g ~ are the unique continuous extensions from BX and BY into BT given by Theorem 6.2, and g~ I Y = g. Now g~(Y) c T so f~(Y) c T which in turn implies Y c (f~rl (T). Therefore, by Theorem 6.23, Y c uX. Then by Lemma 6.17, uY = uX. But since Y is topologically complete, Y = uX.-
EXERCISES 1. If I Y, F) is an inverse limit system where Y = I Y a Ia E A} and F = {~:Y a -7 Y ~ Ia,B E A with B < a} and if for each a E A, X a C Y a , then the inverse limit of {X,G} is a subset of the inverse limit of {y, F} where X = {X a I a E A} and G
= {~IXal a,B
E
A with B < a}.
6.7 The Tamano-Morita Paracompactification
197
2. Show that uX is characterized as a topologically complete space Y which is the smallest with respect to properties (1) and (2) below: (1) Y contains X as a dense subspace. (2) Every real-valued bounded continuous function on X can be extended to a continuous function over Y. 3. Show that if X is the topological sum of IX A IA topological sum of I uX A I A E A}.
E
A}, then uX is the
6.7 The Tamano-Morita Paracompactification As an example of the Tamano-Morita paracompactification of a space X, we consider the case where X is an M-space. This example was given by K. Morita in his paper Topological completions and M-spaces referenced in Section 5.5, as an example of a space whose topological completion is paracompact. M-spaces were introduced in 1964 by Morita in a paper titled Products of Normal Spaces with Metric Spaces (Math. Annelen, Volume 154, pp. 365-382). Paracompact M-spaces, unlike paracompact spaces, are countably productive. M-spaces are defined as follows: a space X is said to be an M-space if there exists a normal sequence {Un} of open coverings of X satisfying the condition (M) below: (M)
If IKn} is a decreasing sequence of non-empty closed sets in X such that Kn C S(x, Un) for each positive integer n, and for some x E X, then nKn 0.
*
Let X be an M-space and let {$A I A E A'I be the collection of all normal sequences of open coverings of X satisfying condition (M). It is left as an exercise to show that A' is cofinal in A. Hence by Exercise 1 of Section 6.6, uX is the inverse limit of the inverse limit subsystem IXN.lI,., ~ IA E A'I of IX/CPA' ~ IA E AI. Recall that a continuous function f:X ~ Y is said to be a perfect mapping if f is closed and for each p E Y, 1 (P) is a compact subset of X. f is said to be quasi-perfect if f is closed and 1 (p) is countably compact for each pE Y.
r
r
LEMMA 6.19 For each A E A', A:X ~ X/CPA is a quasi-peifect mapping. Proof: Let A E A' and let CPA = {Un I. Then A:X ~ X/CPA is the mapping defined in Section 5.5 by <1>1.. = ~ © i A where i A:X ~ X A is the identity mapping on X viewed as a mapping from X onto X A and ~ is the quotient mapping from X A onto X/CPA' X A is the pseudo-metric space (X, d A) where d A is the pseudo-metric generated by ct>A' From the proof of Theorem 6.23, ~ is an open mapping. To show that <1>1.. is closed, let F be a closed subset of X and
198
let y* E CI«h.(F». Let x we have:
6. Paracompactifications E
<1>~I(y*).
Since Star(5;(x, U n+l ), Un+d c sex, Un),
Hence V = <1>~(/n{XA (S(x. Un))) is open in Xj
r
1(F) is closed in X so (g © /)(/-1 (F» = g(/(/-l (F») = g(F) is closed in Z. Therefore, g is a closed mapping. Next let z E Z. Then (g © I rl (z) is countably compact in X which implies I«g © f)-I (z» is countably compact in Y. But I«g © f)l (z» =1(/-1 (g-l (z») = rl (z) so g is a quasi-perfect mapping. -
Proof: Let F be closed in Y. Then
As a consequence of Lemmas 6.18 and 6.19, if "A., j..l E A' with "A. < j..l, we have <1>A = <1>~ © <1>)1; and since <1>A and <1>)1 are quasi-perfect, we must have that <1>~ is quasi-perfect. Let (<1>~)~ denote the Stone extension of <1>~. Then (<1>~)~(X/ct>)1) = X/
This fact allows us to establish the following theorem: THEOREM 6.25 (K. Morita, 1970) II X is a Tychonojf M-space then uX = (/~)-l (T)for any quasi-perfect mappingffrom X onto a metric space T.
6.7 The Tamano-Morita Paracompactification
199
Proof: The proof consists in showing that a quasi-perfect mapping f from X onto a metric space T coincides with ,,:X ~ X/cD" for some A E N. From the proof of Theorem 6.23, we already know that f = g © <1>" for some A E A and for some continuous g:X/cD" ~ T defined by g(x*) = fez) where z E x*. Let! Un l and! Vn l be as defined in the proof of Theorem 6.23. Clearly g is onto since for l (t) which implies g(x*) = [(x) = t. To show g each t E T we can pick an x E is one-to-one, let x* and y* be distinct elements of X/cD". Then there exists a positive integer n such that S(x,Vn)nS(y,V n) = 0 for some x E x* and y E y*. Therefore, S(f(x) , Un)nS(f(y), Un) = 0. Since g(x*) = j(x) and g(y*) = fey), g(x*) g(y*) so g is one-to-one.
r
"*
Finally, to show g is open, let U be open in X/cD" and let x* E U. Then there exists a positi vc integer n such that S(x*, 2 -n) C U so g(S(x*, 2- n c g(U) and contains g(x*). Therefore, if g(S(x*, 2- n» is open in T then x* is an interior point of g(U). Since x* was chosen arbitrarily, g(U) is open. Let x E x*. Then
»
g(S(x*,2- n»
=
UE TIY*E S(x*,2- n),YE y*,andf(y)=tl =
Now X - SexY n) is closed so f(X - SexY n» = T - S(f(x), Un) is closed in T since f is a closed mapping. Therefore, S(f(x) , Un) is open in T so g is an open mapping. Hence g is a homeomorphism so we can identify T with X/cD" which implies f coincides with <1>". It remains to show that A E N. We do this by showing that {V n} satisfies (M). Let K = {Kn} be a decreasing collection of closed sets in X such that for
some p E X, Kn c S(P,V n) for each positive integer n. Put q =f(P). Thenf(K n) c Seq, Un-I) for each n since Kn c S(p,V n). Now the f(Kn) are closed subsets of T and {f(Kn)} is a decreasing sequence so q E f(Kn) for each positive integer n. Therefore, l (q)nK n 0 for each n. Hence {f-I (q)nKn l is a decreasing 1 (q) sequence of closed sets in X. Since is countably compact, I'In[j-l (q)nKnl 0 so nKn 0. Therefore, {V n l satisfies (M). •
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"*
"*
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"*
LEMMA 6.21 Let f:X ~ T where f is continuous and T is metric. If X is an M-space then :uX ~ T and the following assertions hold: ( J ) f is onto if and only if is onto. (2) f is closed if and only if is closed and (3) f is quasi-perfect if and only if is perfect.
r
r
r
r
Proof: From the proof of Theorem 6.23, we know there exists a continuous function g:X/cD" ~ T such that f = g © <1>" for some A E A. The proof can be modified to show that A can be chosen in such a way that A E N. For this let {Un} and {V n} be as in the proof of Theorem 6.23. Then cD" = {V n }. Pick J..1. E N such that A < J..1. and let cDll = {W n ). Then
200
6. Paracompactifications
positive integer n, Y2 E S(y 1,W n). Since A < ~, Y2 E S(y t>V n)for each positive integer n which implies f(Y2) E S(f(y d,U n ) for each n. Therefore, fey 1) = fey 2)' Define h: X/c.l>" ~ T by h(y*) = fey 1) where Y I E y*. Then f = h ©
X
f
T
~
t~ uX
~
t
X/c.l>"
Clearly if f is onto then r is onto. If r is onto then h is onto since r = g u ©
f is closed and let F be closed in
uX. Put H =
FnX. Then f(H) is closed in T which implies h- l (f(H» is closed in X/c.l>" which in tum implies u«
r
To prove (3), assume f is quasi-perfect. Then by (2) above, is closed. If p E uT then p E ~T which implies (f~rl (P) is closed and hence compact in ~X. Since X is an M-space, by Theorem 6.25, uX = (f~rl (T) so (furl (P) = (f~rl (P). Hence (furl (P) is compact so is perfect. Conversely, assume is perfect. By (2) above, f is closed. Hence, as shown in the proof of Theorem 6.25, h is a homeomorphism of X/c.l>" onto f(X) c T. Identifying X/c.l>" with f(X), we see that f agrees with
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r
THEOREM 6.26 If X is a Tychonoff M-space, then X is topologically preparacompact and the Tamano-Morita paracompactification of X is (f~rl (T) for any quasi-perfect mapping from X onto a metric space T. Proof: Let A E N. Then
6.7 The Tamano-Morita Paracompactification
201
EXERCISES 1. Let {
~
Y be a perfect mapping. Show that if Y is countably compact then
Chapter 7
REALCOMP ACTIFICATIONS
7.1 Introduction From Stone's characterization of ~X (Theorem 6.2), it can be seen that if! is a real valued bounded continuous function on X, then! can be uniquely extended over ~X. Consequently, any continuous function !:X -7 [0,1] has a unique continuous extension over ~X. In fact, ~X can be characterized by this property. To see this we need only establish that if each continuous function f:X -7 [0,1] has a unique continuous extension over ~X, then any continuous function !:X -7 Y where Y is a compact Hausdorff space has a continuous extension over ~X. For this let g:X -7 Y be a continuous function from X into a compact Hausdorff space Y. Let F be the collection of all continuous functions from Y into [0,1] and for each! E F put If = [0,1]. Let P = [lfeF!f and define the function <1>:Y -7 P by <1>Cy) = [lfeFfCy) E P. Then <1> is a continuous function so <1> © g:X -7 P is a continuous function. Now for each! E F,f © g:X -7 [0,1] can be uniquely extended to a continuous function ht=~X -7 [0,1]. Define h:~X -7 Pby h(P)
= ( ... ,ht
Then h is also continuous. Let {xa} be a net in X that converges to p. Then { hex a)} is a net in P that converges to h(P). Also, {g(x a)} is a net in Y that converges to some Yp E Y. Hence {<1> © g(x a)} is a net in P that converges to <1>Cyp)' Since Xa E X for each a, and since hf is an extension of!© g,f© g(x a ) = hfix a ) for each a, so {h(xa)} = {<1> © g(xa)}· Consequently, {<1> © g(x a )} converges to h(P) which means <1>Cyp) = h(P). Define y:~X -7 Y by Y(P) = Yp for eachp E ~X. To show that y is a continuous extension of g, suppose x E X. Then clearly Therefore, y is an extension of g over ~X. To show that y is continuous, first notice that <1>:Y -7 <1>(Y) c P is a homeomorphism. This is because the family F separates points and can distinguish points from closed sets [see Lemma 1.6]. Then Y(P) = Yp = <1>-1 (h(P» for each p E ~X so Y= <1>-1 © h. Therefore, y is continuous. Finally, to show that y is unique, suppose B:~X -7 Y is another continuous extension of g over ~X. By Corollary 2.3, both y
Yx
= g(x) so )'(x) = g(x).
7.2 Realcompact Spaces
203
and 8 are uniformly continuous with respect to W(the uniformity of ~X). Byan argument similar to the proof of Theorem 6.2, g:X ---7 Y is also uniformly continuous with respect to~. Since ~X is the completion of (X, ~), by Theorem 4.9, g has a unique uniformly continuous extension over ~X. Therefore, y = O. We record this result as: THEOREM 7.1 ~X is characterized among compactifications of X by the property that each continuousfunctionf:X ---7 [0,1] has a unique continuous extensionfl3:~X ---7 [0,1].
Since ~X is the only compactification of X for which each real valued bounded continuous function can be continuously extended, it is natural to inquire about the nature of spaces Y such that X is dense in Y and every real valued (not necessarily bounded) continuous function on X has a continuous extension over Y. This investigation leads to the concept of a real compact space. In a 1948 paper titled Rings of real valued continuous functions 1 (Transactions of the American Mathematical Society, Volume 64, pp. 45-99), E. Hewitt introduced the concept of a Q-space. Q-spaces are defined as follows: let Z(X) denote the zero sets of real valued continuous functions on the Tychonoff space X [cf. Section 6.2]. A non-empty subfamily Z of Z(X) is said to be CZ-maximal if Z contains no countable subset with empty intersection and Z is maximal with respect to this property. A Q-space is a Tychonoff space such that the intersection of any CZ-maximal family is non-void. This definition of a Q-space is due to Shirota. It is equivalent to Hewitt's original definition [see Definition 13 and Theorem 50, p. 85 of Hewitt's paper]. Shirota, in his 1952 paper A Class of Topological Spaces (referenced in Section 3.2), showed the equivalence between Q-spaces and Tychonoff spaces that are complete with respect to their e uniformity which he called e-complete spaces.
7.2 Realcompact Spaces Today, Q-spaces and e-complete spaces are known as realcompact spaces, a term introduced by L. Gillman and M. Jerison in their book Rings of Continuous Functions published by Van Nostrand in 1960. Hewitt showed that realcompact spaces are precisely the closed subspaces of Cartesian products of real lines. He also showed that for each Tychonoff space X, there exists a realcompact space uX, called the Hewitt realcompactification of X such that X c uX c ~X and uX has the properties: (1) uX is the largest subspace of ~X containing X such that each real valued continuous function can be extended to uX and (2) uX is the smallest realcompact space between X and ~X (i.e., X is realcompact if and only if X = uX). Further, Hewitt showed that uX is completely determined (up to homeomorphism) by the following properties: (1) uX is realcompact, (2) uX contains X as a dense subspace, and (3) every real valued
204
7. Realcompactifications
continuous function on X can be continuously extended to uX. Hewitt proved parts (2) and (3) of the next theorem. The proof given below is due to Shirota.
THEOREM 7.2 (E. Hewitt, 1948 and T. Shirota, 1952) For a Tychonoff space X, the following conditions are equivalent: (1) X is e-complete, (2) X is realcompact, (3) X is homeomorphic to a closed subset of a Cartesian product of real lines with the usual topology. ~ (2). Let (X, e) be complete and let Z be a CZ-maximal family of X. Let U = {Un} be a countable normal covering of X. We first show that there exists a Z E Z and a U E U such that Z c U. As shown in the proof of Lemma 3.5, we can construct a precise closed refinement {Fn} of {Un}, and then use Corollary 2.2 and Proposition 1.4 to construct real valued continuous functions fn:X ~ [0,1] such thatfn(Fn) = 1 andfn(X - Un) = O. For each n, the set Zn = {x E X Ifn(x) ~ 1/2} can be shown to be a zero set and Fn c Zn C Un. If for each positive integer n, Zn does not belong to Z then for each n there exists a countable collection {Z;:') c Z such that Znn[n;;;,n=IZ;:.J = 0. Then n;;;,n=IZ;:' = [n;;;,n=1 Z;:']nX = [n;;;,n=1 Z;:']n[ U;;'=I Zn] = U;;'=I (Znn[n;;;,n=1 Z;:.]) = 0. But this contradicts the assumption that Z is a CZ-maximal family. Hence for some positive integer n, Zn E Z. Then Zn is the desired Z CUE U.
Proof: (1)
If Z I, Z2 E Z then Z 1 nZ2 "# 0 and if Z 1 is the zero-set of f and Z2 is the zero-set of g, then Z 1 nZ2 is the zero-set of fg. Clearly Z 1 nZ2 E Z. Since Zu{Z 1 nZ2) contains Z and satisfies properties (1) through (3) of the definition of a CZ-maximaI family. Let E = {(Z, x) IZ E Z and x E Z) and define <s: on E by (Z I, x) <S: (Z2,y) if Z2 c Z I. Then (E, <S:) is a directed set. For each (Z, x) E E put \jI(Z, x) = x. Then \jI:E ~ X is a net. If U E e, there is a Z E Z and aU E U with Z c U. Then \jI(Z, x) E U for each x E Z and if (Z',y) E E with (Z, x) <S: (Z',y), then Z' c Z so \jI(Z',y) E U. Therefore, \jI is Cauchy with respect to e and hence converges to some p E X. Let Z E Z and let U(P) be an open set containing p. Since \jI is eventually in U(P), there is a (Z', x) E E with Z' c Z and \jI(Z', x) = x E U(P), Then U(p)nZ "# 0 so P E Clx(Z) = Z, Hence p E n {Z IZ E Z). Therefore, X satisfies condition (2). This completes the proof of (1)
~
(2).
To show that (2) ~ (3), let C denote the family of all real valued continuous functions on X and for each f E C let Rf = R. Set P = TIfEcRf and let h:X ~ P be the product mapping defined in Section 1.3 by [h (x)]f = f(x). Since X is Tychonoff, the family C can distinguish points and can distinguish points from closed sets. Consequently, by Lemma 1.6, h is an imbedding, so heX) is homeomorphic with X. We may therefore assume X c P. Let 1t be the product uniformity on P and let (X, 1t) denote the uniform subspace of (P, 1t). It remains to show that X is closed in P. By Exercise 3 of Section 4.4, it will suffice to show that (X, 1t) is complete.
7.2 Realcompact Spaces
205
For this let {x a } be a Cauchy net in (X, 1t'). For each a put H a = Cl{x~ Ia E 1t, there is an a such that Hac U for some U E U. Moreover, since there exists a continuous function f:X """"* [0,1) with f(H a) = 0 and f(X - U) = 1, we have for the zero-set Z(j) that Hac Z(j) C U. Let Z' be the family of zero-sets of X that contain some H a' Then if Z 1, Z2 E Z', Z1nZ2 E Z'. By Zorn's Lemma (Theorem 0.1(4», it is possible to find a maximal subfamily Z of Z(X) with respect to the finite intersection property that contains Z'. We now show that Z is also CZ-maximal.
< ~}. It is easily shown that for each U
Suppose, on the contrary, there is a countable subfamily {Z;} of Z with ni'=IZi = 0. Then for each positive integer i, there is anj; E C with Zi = Z(fi). In fact, we can assume that for each i, fi:X """"* [0,1]. Now for each positive integer n, define gn:X """"* [0,1] by gn(x) = maxI Ij;(x) II i = 1 ... n} for each x E X. Put g = L;;'=12-ngn' Then since ni'=IZi = 0, we have g(x) > 0 for each x E X. Consequently, g-1 defined by g-I(X) = l/g(x) is continuous. Therefore, g-1 E C. Since 1t is the product uniformity on P, 1t is the coarsest uniformity on P that makes all the canonical projections uniformly continuous. But for each f E C, Pt = f, so g-1 is uniformly continuous. Since for each U E 1t, there is a Z E Z with Z c U for some U E U, and since g -1 is uniformly continuous, then for each £ > 0, there is a Z E Z such that g-1 (Z) is contained in a set of diameter £. Now if x E Z(gn), then g(x) < 2- n+1. Hence there exists a sufficiently large n with g-I(X) > max{g-I(y)ly E Z} for any x E Z(gn)' This implies that Z(gn)nZ = 0. But Z(gn) = n7=1 Z(fi) which implies that Z(gn) E Z. Then Z does not satisfy the finite intersection property which is a contradiction. Therefore, Z is a CZ-maximal family. Consequently, by property (2), Z has a non-empty intersection. Let pEn {Z IZ E Z}. Let U E 1t' and choose V E 1t' such that V* < U. Suppose no Z E Z meets S(P,V). Since there exists a Z(V) E Z and a V E V with Z(V) c V, we see that p does not belong to Z(V) which is a contradiction. Therefore, Z(V)nS(p,V) "# 0 which implies there is an open neighborhood V' of p such that V'nV"# 0 so there is aU E U with V\..;V cU. But then Z(V) c S(p, U) and Z(V) contains H a for some index a. Hence, {x a} converges to p so (X, 1t') is complete. This completes the proof of (2) """"* (3). To show (3) """"* (1), let X be a closed subset of a product P = IlaR a where R a = R for each index a. Since R a is complete for each a, so is P. Let 1t denote the product uniformity on P and let 1t' denote 1t restricted to X. Since X is closed in P, (X, 1t) is complete. It will suffice to show that e is a basis for 1t'. For this let U be a basic member of 1t. Then
for some finite collection of uniform coverings Ua, of R a, where P a, denotes the canonical projection of Ponto Raj for each i = 1 ... n. Now each Uaj has a countable subcovering Vaj since Raj is LindelOf. Furthermore, each Va, is normal since Raj is paracompact. Therefore, P~~ (Val)n ... npu~(Va") is a
206
7. Reaicompactifications
countable normal covering of P that refines U. Hence U is a member of the e uniformity for P. But then e is a basis for the n' uniformity on X. Therefore, e =n', so X is e-complete. Now that we know realcompact spaces are precisely the e-complete spaces, we can use this fact to derive some interesting properties about realcompact spaces.
THEOREM 7.3 Every Lindel6f space is reaicompact. This is because Lindelbf spaces are cofinally complete with respect to the e uniformity by Theorem 4.4 and hence complete with respect to e hy Corollary 4.1.
THEOREM 7.4 A closed subspace of a reaicompact space is reaicompact. This is because if Y is a closed subspace of a realcompact space X, then Y is complete with respect to the e uniformity of X relativized to Y by Proposition 4.18. But the e uniformity of X relativized to Y is precisely the e uniformity of Y.
THEOREM 7.5 A product of reaicompact spaces is reaicompact. Proof: Let X = flaX a where each X a is realcompact and for each a, let e a be the e uniformity of X a' Then for each a, (X a' e a) is complete so X is complete with respect to the product uniformity by Exercise 4 of Section 5.3. Let /-l denote the product uniformity of the e a's. Then a basis element of /-l is of the form
for some finite collection of uniform coverings U a , of e a, where each U a , is countable and where p a, denotes the canonical projection of X onto X a, for each i = 1 ... n. Then U is also countable and therefore belongs to e (the e uniformity of X). Hence /-l c e. Since (X, /-l) is complete and e is finer than /-l, we have that (X, e) is complete so X is realcompact.We have already seen in Sections 2.2 and 5.5 how a normal sequence of open coverings can be used to construct a pseudo-metric. If we consider the family
£
> 0 there exists a d'
E
D
7.2 Realcompact Spaces
207
and a 8 > 0 such that d'(x,y) < 8 implies d(x,y) < e for all x,y E X, then d ED, (3) whenever x -:/- y, there exists a d E D with d(x,y) -:/- O. where dye is defined by dve(x,y) = min {d(x,y), e(x,y)} and is called the join of d and e. Property (3) only holds for Hausdorff uniform spaces and property (2) implies that if dE D, then rd E D for each r> 0; and if dE D and e :::; d then e E
D. To show property (l) holds let d 'J..., deE D and let 'J... and