(2) A spaceX is metrizable if it is a locally finite union of metrizable closed subspaces.
Sketch of Proof.To apply (1) above, construct a surjective perfect map f:λ∈ΛXλ→Xsuch that eachXλis metrizable andf|Xλis a closed embedding. The metrizability ofλ∈ΛXλeasily follows from Theorem 1.3.4. (The metrizability ofλ∈ΛXλcan also be seen by embedding
λ∈ΛXλinto the product spaceΛ×ℓ1(Γ)for someΓ, where we give Λthe discrete topology.)
1.5 Complete Metrizability 41
y0 =x ε0=ε
ε1
y1
B(y0, ε0)
B(y1, ε1)
Fig. 1.5.Definition ofyn∈X andεn>0
1.5.2 Theorem A metrizable space is completely metrizable if and only if it is absolutelyGδ.
This follows from Corollary 1.3.6 (or 1.3.10) and the following theorem:
1.5.3 Theorem LetX= (X, d)be a metric space andA⊂X. (1) IfA is completely metrizable, thenAisGδ inX.
(2) IfX is complete andAisGδ inX, thenAis completely metrizable.
Proof. (1) Since clAisGδ in X, it suffices to show that AisGδ in clA. Let ρ∈Metr(A) be a complete metric. For eachn∈N, let
Gn=
x∈clAxhas a neighborhoodU in X with diamdU <2−n and diamρU∩A <2−n
. Then, eachGn is clearly open in clA andA⊂
n∈NGn. Eachx∈
n∈NGn
has neighborhoods U1 ⊃ U2 ⊃ · · · in X such that diamdUn < 2−n and diamρUn∩A < 2−n. Since x ∈ clA, we have points xn ∈ Un ∩A, n ∈ N.
Then, (xn)n∈N converges tox. Since (xn)n∈Nisρ-Cauchy, it is convergent in A. Thus, we can conclude that x∈A. Therefore,A=
n∈NGn, which isGδ
in clA.
(2) First, we show that any open setU inX is completely metrizable. We can define an admissible metricρforU as follows:
ρ(x, y) =d(x, y) +d(x, X\U)−1−d(y, X\U)−1.
Every ρ-Cauchy sequence (xn)n∈N in U isd-Cauchy, so it converges to some x∈X. Since (d(xn, X\U)−1)n∈Nis a Cauchy sequence inR, it is bounded.
Then,
d(x, X\U) = lim
n→∞d(xn, X\U)>0.
This means that x ∈ U, and hence (xn)n∈N is convergent in U. Thus, ρ is complete.
Next, we show that an arbitraryGδ set Ain X is completely metrizable.
WriteA=
n∈NUn, whereU1, U2, . . . are open inX. As we saw above, each Un admits a complete metric dn ∈ Metr(Un). Now, we can define a metric ρ∈Metr(A) as follows:
ρ(x, y) =
n∈N
min
2−n, dn(x, y) .
Everyρ-Cauchy sequence inAisdn-Cauchy, which is convergent inUn. Hence, it is convergent inA=
n∈NUn. Therefore,ρis complete. ⊓⊔
Analogous to compactness, the completeness of metric spaces can be char-acterized by the finite intersection property (f.i.p.).
1.5.4 Theorem In order for a metric space X = (X, d)to be complete, it is necessary and sufficient that, if a family F of subsets ofX has the finite intersection property and contains sets with arbitrarily small diameter, then Fcl has a non-empty intersection, which is a singleton.
Proof. (Necessity) LetFbe a family of subsets ofXwith the f.i.p. such thatF contains sets with arbitrarily small diameter. For eachn∈N, chooseFn∈ F so that diamFn < 2−n, and take xn ∈ Fn. For anyn < m, Fn ∩Fm = ∅, hence
d(xn, xm)diamFn+ diamFm<2−n+ 2−m<2−n+1.
Thus, (xn)n∈Nis a Cauchy sequence, therefore it converges to a pointx∈X.
Then,x∈
Fcl. Otherwise,x∈clF for someF ∈ F. Choosen∈Nso that d(x, xn), 2−n< 12d(x, F). SinceF∩Fn=∅, it follows that
d(x, F)d(x, xn) + diamFn< d(x, xn) + 2−n< d(x, F), which is a contradiction.
(Sufficiency) Let (xn)n∈N be a Cauchy sequence in X. For each n ∈ N, let Fn = {xi | i n}. Then, F1 ⊃ F2 ⊃ · · · and diamFn → 0 (n → ∞).
From this condition, we have x∈
n∈NclFn. For each ε >0, choosen∈ N so that diam clFn = diamFn < ε. Then, d(xi, x) < ε for i n, that is, limn→∞xn=x. Therefore, X is complete. ⊓⊔
Using compactifications, we can characterize complete metrizability as fol-lows:
1.5.5 Theorem For a metrizable spaceX, the following are equivalent:
(a) X is completely metrizable;
(b) X isGδ in an arbitrary compactification ofX; (c) X isGδ in the Stone– ˇCech compactificationβX;
1.5 Complete Metrizability 43 (d) X has a compactification in whichX isGδ.
Proof. The implications (b)⇒(c)⇒(d) are obvious. We show the converse (d)⇒(c)⇒(b) and the equivalence (a)⇔(b).
(d)⇒(c): LetγX be a compactification ofX and X =
n∈NGn, where eachGnis open inγX. Then, by Theorem 1.1.4, we have a mapf :βX→γX such that f|X = id, where X = f−1(X) by Theorem 1.1.8. Consequently, X =
n∈Nf−1(Gn) isGδ in βX.
(c) ⇒ (b): By condition (c), we can write βX\X =
n∈NFn, where eachFn is closed in βX. For any compactification γX of X, we have a map f : βX → γX such that f|X = id (Theorem 1.1.4). From Theorem 1.1.8, γX\X =f(βX\X) =
n∈Nf(Fn) isFσ in γX, henceX isGδ in γX.
(b)⇒ (a): To prove the complete metrizability ofX, we show thatX is absolutelyGδ (Theorem 1.5.2). LetX be contained in a metrizable spaceY. Since clβY X is a compactification of X, it follows from (b) thatX is Gδ in clβY X, and hence it isGδ in Y ∩clβY X = clY X, where clY X is alsoGδ in Y. Therefore,X isGδ in Y.
(a)⇒(b): LetγXbe a compactification ofXanddan admissible complete metric for X. For each n∈ N and x∈ X, letGn(x) be an open set inγX such thatGn(x)∩X = Bd(x,2−n). Then,Gn =
x∈XGn(x) is open inγX and X ⊂Gn. We will show that each y ∈
n∈NGn is contained inX. This implies thatX =
n∈NGn isGδ inγX.
For eachn∈N, choosexn∈X so thaty∈Gn(xn). Sincey∈clγXX and Gn(xn)∩X = Bd(xn,2−n), it follows that{Bd(xn,2−n)|n∈N}has the f.i.p.
By Theorem 1.5.4, we havex∈
n∈NclXBd(xn,2−n), where limn→∞xn=x because d(xn, x)2−n. Thus, we have y =x∈ X. Otherwise, there would be disjoint open sets U and V in γX such that x ∈ U and y ∈ V. Since y∈
n∈NGn∩V,{Bd(xn,2−n)∩V |n∈N}has the f.i.p. Again, by Theorem 1.5.4, we have
x′ ∈
n∈N
clX(Bd(xn,2−n)∩V)⊂clXV.
Since limn→∞xn=x′ is the same asx, it follows thatx′=x∈U, which is a contradiction. ⊓⊔
Note that conditions (b), (c), and (d) in Theorem 1.5.5 are equivalent without the metrizability of X, but X should be assumed to be Tychonoff in order that X has a compactification. A Tychonoff space X is said to be Cech-completeˇ ifX satisfies one of these conditions.
Every compact metric space is complete. Since a non-compact locally com-pact metrizable space X is open in the one-point compactification αX = X ∪ {∞}, X is completely metrizable because of Theorem 1.5.5. Thus, we have the following corollary:
1.5.6 Corollary Every locally compact metrizable space is completely met-rizable. ⊓⊔
We now state and prove theLavrentieffGδ-Extension Theorem:
1.5.7 Theorem (Lavrentieff) Let f :A→Y be a map from a subsetA of a spaceX to a completely metrizable spaceY. Then,f extends over aGδ
set Gin X such thatA⊂G⊂clA.
Proof. We may assume thatY is a complete metric space. The oscillation of f atx∈clA is defined as follows:
oscf(x) = inf
diamf(A∩U)U is an open neighborhood ofx . Let G = {x ∈ clA | oscf(x) = 0}. Then, A ⊂ G because f is continuous.
Since each{x∈clA|oscf(x)<1/n}is open in clA, it follows that GisGδ
in X. For eachx∈G, Fx=
f(A∩U)U is an open neighborhood ofx ,
has the f.i.p. and contains sets with arbitrarily small diameter. By Theorem 1.5.4, we have
Fxcl =∅, which is a singleton because diam
Fxcl = 0. The desired extension ˜f :G→Y off can be defined by ˜f(x)∈
Fxcl. ⊓⊔ IfAis a subspace of a metric spaceX andY is a complete metric space, then every uniformly continuous map f : A → Y extends over clA. This result can be obtained by showing thatG= clAin the above proof. However, a direct proof is easier.
We will modify Theorem 1.5.7 into the following, known as the Lavren-tieff Homeomorphism Extension Theorem:
1.5.8 Theorem (Lavrentieff) Let X and Y be completely metrizable spaces and letf :A→B be a homeomorphism betweenA⊂X and B⊂Y. Then, f extends to a homeomorphismf˜:G→H betweenGδ sets inX and Y such thatA⊂G⊂clAand B⊂H ⊂clB.
Proof. By Theorem 1.5.7, f and f−1 extend to maps g : G′ → Y and h : H′ →X, whereA⊂G′ ⊂clA,B ⊂H′ ⊂clB andG′,H′ areGδ in X and Y, respectively. Then, we haveGδ sets G=g−1(H′) and H =h−1(G′) that contain Aand B as dense subsets, respectively. Consider the maps h(g|G) : G→X andg(h|H) :H →Y. Sinceh(g|G)|A= idA and g(h|H)|B= idB, it follows thath(g|G) = idG andg(h|H) = idH. Then, as is easily observed, we haveg(G)⊂Handh(H)⊂G. Hence, ˜f =g|G:G→H is a homeomorphism extendingf. ⊓⊔
In the above, whenX =Y andA =B, we can take G=H, that is, we can show the following:
1.5.9 Corollary LetX be a completely metrizable space andA⊂X. Then, every homeomorphism f : A→A extends to a homeomorphism f˜:G→G over aGδ setGinX withA⊂G⊂clA.