3.5 Fr´ echet Spaces and Countable Products of ARs
In this section, it is shown that every Fr´echet space is homeomorphic to a Hilbert space with the same weight, and that the countable product of com-pletely metrizable ARs is homeomorphic to a Hilbert space under some condi-tion. We also modify the conditions characterizingℓ2(Γ)-manifolds (orℓ2(Γ)).
The following is called theuniformly separating cells property:
(USC) For eachε >0, there existsδ >0 such that any mapf :In→X(n∈ N) isε-homotopic to a mapg:In→X with dist(f(In), g(In))> δ.
First, we show the following:
Lemma 3.5.1 For each path-connected metric space X = (X, d), the uni-formly separating cells property implies the discrete cells tower property, and hence the locally finite cells tower property.
Proof. We may assume that d is bounded. Denote D = ⊕
n∈NIn and let f :D→X andα:X →(0,1) be maps. For eachk∈N, let
Dk ={
x∈Dαf(x)>3−k} .
We construct sequences gk : D → X and εk > 0, k ∈ N, so that for each k∈N,
(1) gk|D\Dk+1=f|D\Dk+1 andgk|Dk−1=gk−1|Dk−1; (2) dist(gk(In∩Dk), gk(Im))> εk ifm < n;
(3) d(gk, gk−1)< 13εk−1; (4) εk6 1
3εk−1,
whereε0= 3−2, g0=f andD0=∅. Note thatεk 63−k−2 for eachk∈N. Assumegk−1 andεk−1 have been constructed. Choose a neighborhoodU ofDk−1 in Dso that
(5) dist(gk−1(In∩U), gk−1(Im))> εk−1ifm < n.
Forε=13εk−1>0, letδ >0 be as (USC) and defineεk = min{ε, δ}6 13εk−1. Then, (4) is obvious. By induction, we define a map gk|In as follows: Let gk|I1=gk−1|I1. Assume thatgk|I1⊕ · · · ⊕In−1 is defined so as to satisfy (1), (2), and (3). RegardingI⊂I2⊂ · · · ⊂In, we define
F =I× {0} ∪
∪n i=1
{1/i} ×Ii⊂In+1.
By the path-connectedness of X, we have a map v : F → X such that v(1/i, x) = gk(x) for x ∈ Ii, i < n, and v(1/n, x) = gk−1(x) for x ∈ In. Combining with a retraction of In+1 onto F, we can extend v to a map w : In+1 → X. As the restriction of an ε-homotopy obtained by applying (USC) tow, we have anε-homotopys:In×I→X such thats0=gk−1|In and
180 3 Hilbert Manifolds and Hilbert Cube Manifolds
1/2 1 1/3
I I2 I3 In
1/n
In+1
Fig. 3.3. I× {0} ∪
∪n i=1
{1/i} ×Ii
(6) dist(s1(In), gk(I1⊕ · · · ⊕In−1))> δ.
Recall thatDk∩cl(D\Dk+1) =∅andDk−1⊂U. Letλ:D→Ibe a Urysohn map with
λ(Dk\U) = 1 and λ(Dk−1∪cl(D\Dk+1)) = 0.
We definegk|In:In →X by
gk(x) =s(x, λ(x)) for eachx∈In.
Then,gk|I1⊕ · · · ⊕In satisfies (1) and (3). To see (2), letm < n,x∈In∩Dk, and y ∈ Im. If x ̸∈ U, then gk(x) = s1(x), hence it follows from (6) that d(gk(x), gk(y))> δ > εk. If x∈U, then d(gk−1(x), gk−1(y))> εk−1 by (5), d(gk(x), gk−1(x)) < 13εk−1, and d(gk(y), gk−1(y)) < 13εk−1 by (3), hence it follows that d(gk(x), gk(y))> 13εk−1>εk.
By (1) and (3), we have a map g : D → X as the limit of gk, that is, g|Dk = gk|Dk for each k ∈ N. For each x ∈ D, choose k ∈ N so that x∈Dk+1\Dk. Then, by the definition ofg, (1), and (3),
d(g(x), f(x)) =d(gk+1(x), gk−1(x))< 13εk+13εk−1
<12εk−1<123−k−1<3−k−16αf(x).
We shall show that (g(In))n∈N is discrete in X. It follows from (2) that (g(In))n∈N is disjoint. Then, it suffices to show that (g(In))n∈N is locally finite in X. Assume that there is a sequence (g(xi))i∈N convergent toy ∈X such that distinct xi’s belong to distinct cells inD. Then, infi∈Nαf(xi)>0, otherwise (f(xi))i∈Ncontains a subsequence (yi)i∈Nsuch thatα(yi)→0 and yi →y, which contradictsα(y)>0. Therefore, we can choosek∈Nso that xi∈Dk for alli∈N. Then, by (2),
d(g(xi), g(xj)) =d(gk(xi), gk(xj))> εk if i̸=j.
3.5 Fr´echet Spaces and Countable Products of ARs 181 This is contrary to the assumption. Thus, it follows that (g(In))n∈Nis locally finite inX. ⊓⊔
We can apply Lemma 3.5.1 above to topological groups.
Proposition 3.5.2 Let G be a locally path-connected metrizable topological group. If any neighborhood of the unit e∈G is not totally bounded for some admissible right invariant metric d then (G, d) has the uniformly separating cells property, henceGhas the locally finite cells tower property.
Proof. Observe that each path-component ofGis clopen inGand is homeo-mophic to the component of the unit e∈G that is a clopen subgroup ofG.
Hence, we may assume thatGis connected, and hence is path-connected.
For eachε >0, chooseη >0 so that d(x, e)< η implies thatxandecan be connected by a path inGwith diam< ε. Also chooseδ >0 so that B(e, η) has no finite cover with mesh64δ. For any map f :In→G,
A={
f(x)·f(y)−1x, y∈In}
is compact. Then, we can choosea∈B(e, η) so thatd(a, A)> δ. Letρ:I→G be a path such that ρ(0) =e, ρ(1) =a, and diamρ(I)< ε. We define an ε-homotopyh:In×I→Gby
h(x, t) =ρ(t)·f(x) for each x∈In and t∈I.
Then,h0=f and, for eachx, y∈In,
d(h1(x), f(y)) =d(a, f(y)·f(x)−1)>d(a, A), that is, dist(f(In), h1(In))>d(a, A)> δ. ⊓⊔
A linear topological space is finite-dimensional if and only if0has a totally bounded neighborhood. Then, we have the following:
Corollary 3.5.3 A topological linear space is homeomorphic toℓ2if and only if it is an infinite-dimensional separable completely metrizable AR. ⊓⊔
Since a Fr´echet space (= a locally convex completely metrizable topological linear space) is an AR, we have theKadec–Anderson Theoremas a special case of this corollary.
Theorem 3.5.4 (Kadec–Anderson) Every infinite-dimensional separable Fr´echet space is homeomorphic toℓ2, and hence toRN. ⊓⊔
Next, we apply Lemma 3.5.1 to show that infinite products have the locally finite cells tower property.
182 3 Hilbert Manifolds and Hilbert Cube Manifolds Proposition 3.5.5 Let X = ∏
i∈NXi be the countable product of path-connected completely metrizable spaces. If infinitely many of Xi’s are non-compact thenX has the uniform cells sepalation for some admissible metric, henceX has the locally finite cells tower property.
Proof. For each i ∈ N, letdi ∈ Metr(Xi). We assume thatdi is not totally bounded ifXiis non-compact. Then, each non-compactXi has no finite cover with mesh< εi for some εi >0. We can assume that εi >4. Thus, ifXi is non-compact andA⊂Xi is compact, thendi(x, A)>1 for somex∈Xi. We define a metricdforX as follows:
d(x, y) =∑
i∈N
min{
2−i, di(x(i), y(i))} .
We show that (X, d) has (USC). For each ε > 0, choose k ∈ N so that 2−k < ε/2 and Xk is non-compact. For each map f : In → X and i ∈ N, let fi = prif :In → Xi, where pri :X → Xi is the projection. Since Xk is non-compact, we have a∈Xk with d(a, fk(In))>1. SinceIn is contractible and Xk is path-connected, we have a homotopy h: In×I→ Xk such that h0 = fk and h1(In) = a. We define a homotopy g : In×I → ∏
i∈NXi as follows:
g(x, t) = (f1(x), . . . , fk−1(x), h(x, t), fk+1(x), . . .).
For each x∈In andt∈I,
d(f(x), g(x, t)) = min{
2−k, dk(fk(x), h(x, t))}
< ε/2, that is,gis anε-homotopy. Clearlyg0=f. For eachx, y∈In,
d(g1(x), f(y)) = min{
2−k, dk(a, fk(In))}
= 2−k, that is, dist(f(In), g1(In)) = 2−k. ⊓⊔
As a corollary, we have the following:
Corollary 3.5.6 LetX =∏
i∈NXibe the countable product of separable com-pletely metrizable connected ANRs. If the Xi’s are ARs except for finitely many i∈N (or all Xi’s are AR) and infinitely many Xi’s are non-compact thenX is an ℓ2-manifold (orX ≈ℓ2). ⊓⊔
Now, we treat the non-separable case. We start with the following:
Lemma 3.5.7 Assume that X ≈X ×ℓ2. Let f : ⊕
γ∈Γ Yγ →X be a map such that (f(Yγ))γ∈Γ isσ-discrete (or σ-locally finite) inX. Then, for each U ∈ cov(X), f is U-close to a map g such that (g(Yγ))γ∈Γ is discrete (or locally finite) in X.
3.5 Fr´echet Spaces and Countable Products of ARs 183 Proof. Since ℓ2 ≈ RN by Corollary 3.5.3 or 3.5.6, prX : X ×RN → X is U-close to a homeomorphism h : X ×RN → X by Corollary 2.3.4(iii). We write Γ = ⊕
n∈NΓn, where each (f(Yγ))γ∈Γn is discrete in X. We define f˜:⊕
γ∈ΓYγ →X×RN by ˜f(y) = (f(y), n,0,0, . . .) ify ∈⊕
γ∈ΓnYγ. Then, g=hf˜is the desired map. ⊓⊔
Lemma 3.5.8 LetX be a completely metrizable ANR withw(X) =τ. Then, X is anℓ2(Γ)-manifold ifX≈X×ℓ2andX satisfies the following condition with respect to somed∈Metr(X):
(∗)For anyn∈Nandε >0, eachf :In×Γ →X isε-close to a mapg such that(g(In× {γ}))γ∈Γ isσ-locally finite inX.
Proof. Note that, for any map f : PΓ → X, (f(PΓn))n∈N is σ-discrete (σ-locally finite) inX. Then, X satisfies (LFSτ) by Lemma 3.5.7.
To see thatX satisfies (τ-LFC), letf :In×Γ →X andα:X →(0,1) be maps. By Lemma 3.5.7, it suffices to find a mapg:In×Γ →X such that g∈Nα(f) and (g(In× {γ}))γ∈Γ isσ-locally finite inX. For eachi∈N, let
Γi={
γ∈Γ 2−i6inf{αf(z, γ)|z∈In}<2−i+1} . Then,Γ =⊕
i∈NΓi. By condition (∗), we have mapsgi:In×Γi→X,i∈N, such thatd(gi, f|In×Γi)<2−iand (gi(In× {γ}))γ∈Γiisσ-locally finite inX.
Letg:In×Γ →X be the map defined byg|In×Γi =gi. Then,g is clearly the desired one. ⊓⊔
We use the following lemma to prove the non-separable version of Corollary 3.5.6.
Lemma 3.5.9 Let X = ∏
i∈NXi be the countable product of completely metrizable connected ANRs (or ARs) such that w(X) = τ. If each Xi is an AR containing a closed copy of ℓ2(Γ)except for finitely manyi∈N, then X is anℓ2(Γ)-manifold (or X ≈ℓ2(Γ)).
Proof. It suffices to verifyM1(τ)-universality in Theorem 3.4.3. We may as-sume thatXi is an AR containing a closed copy ofℓ2(Γ) for everyi >1. Let f :Y →X be a map ofY ∈M1(τ) andα:X→(0,1) be a map. We use the same metricdforX as in the proof of Proposition 3.5.5, that is,
d(x, y) =∑
i∈N
min{
2−i, di(x(i), y(i))} ,
wheredi∈Metr(Xi) for eachi∈N. SinceXi contains a closed copy ofℓ2(Γ), we have a closed embeddinghi :Y →Xi for eachi >1. For each i >1, let λi :Xi2×I→Xi be an equi-connecting map forXi, that is,λi(x, y,0) =x, λi(x, y,1) = y, and λi(x, x, t) =x for each x, y ∈ X and t ∈I. We define a mapg:Y →X as follows:
184 3 Hilbert Manifolds and Hilbert Cube Manifolds g(y) = (pr1f(y), . . . ,prif(y),
λi+1(pri+1f(y), hi+1(y),2iα(f(y))−1),
hi+2(y), hi+3(y), . . .) if 2−i< α(f(y))62−i+1.
Then, d(f(y), g(y))< α(f(y)) for eachy∈Y. It remains to show thatg is a closed embedding. Assume thatg(yn)→x(n→ ∞). Lets= infn∈Nα(f(yn)).
If s = 0 then α(f(yni))→ 0 for some n1 < n2 <· · ·, whence f(yni) → x, which meansα(x) = 0. This is a contradiction. Then, we can choosek∈Nso that 2−k< s. For eachn∈N,α(f(yn))>2−k, hence prk+2g(yn) =hk+2(yn).
Since hk+2 is a closed embedding, (yn)n∈N is convergent in Y. Thus, g is a closed embedding. ⊓⊔
Now, we prove the non-separable version of Corollary 3.5.6:
Theorem 3.5.10 Let X = ∏
i∈NXi be the countable product of completely metrizable connected ANRs such that w(X) = τ = supi>nw(Xi) for each n ∈N. If each Xi is an AR except for finitely many i ∈ N (or all Xi’s are ARs), and infinitely manyXi’s are non-compact, thenXis anℓ2(Γ)-manifold (or X≈ℓ2(Γ)).
Proof. It suffices to show that X contains a closed copy of ℓ2(Γ). In fact, considering products of infinitely manyXiinstead ofXn, we can assume that each Xi contains a closed copy of ℓ2(Γ), hence the result follows from the lemma above.
As is well known, every path-connected space is arcwise connected (cf.
[GAGT, Corollary 5.14.7]). . Then, it is easy to see that the productY1×Y2
[5.14.7]
of non-compact path-connected completely metrizable spaces contains a closed copy of the half real lineR+= [0,∞). By Corollary 3.5.6,RN+≈ℓ2. Without loss of generality, we may assume thatw(X) = limi∈Nw(X2i), where everyX2i
is an AR. Then,Y =∏
i∈NX2i is a completely metrizable AR with w(Y) = w(X) andX contains a closed copy ofY ×ℓ2. We show thatY ×ℓ2≈ℓ2(Γ).
To this end, it suffices to verify condition (∗) in Lemma 3.5.8. We use the following metric forY ×ℓ2:
d((y, z),(y′, z′)) =∑
i∈N
min{
2−i, di(y(i), y′(i))}
+∥z−z′∥.
Letn∈Nand f :In×Γ →Y ×ℓ2 be a map. For eachε >0, choosek∈N so that 2−k < ε. Let Z =∏
i>kX2i. Sincew(Z) =w(X) = cardΓ, Z has a σ-discrete open basis B ={Bγ | γ ∈ Γ}. Then, we have a map φ :Γ → Z such thatφ(γ)∈Bγ for eachγ∈Γ. We define a mapg:In×Γ →Y ×ℓ2 as follows:
g(x, γ) = (pk(f(x, γ)), φ(γ), q(f(x, γ)))∈
∏k i=1
X2i×Z×ℓ2=Y ×ℓ2,
3.5 Fr´echet Spaces and Countable Products of ARs 185 wherepk :Y ×ℓ2→∏k
i=1X2i andq:Y ×ℓ2→ℓ2are the projections. Since Bisσ-discrete inZ, it follows thatφ(Γ) isσ-discrete inZ, which implies that (g(In× {γ}))γ∈Γ isσ-discrete inY ×ℓ2. Thus, Y ×ℓ2 satisfies condition (∗) in Lemma 3.5.8, henceY ×ℓ2≈ℓ2(Γ). ⊓⊔
As a special case of Corollary 3.5.6 and Theorem 3.5.10, we have the following:
Corollary 3.5.11 For every infinite setΓ,J(Γ)N≈ℓ2(Γ). ⊓⊔
Proposition 3.5.12 Let X be a completely metrizable ANR with w(X) = τ > ℵ0. Then, X is an ℓ2(Γ)-manifold if and only if X ≈ X ×ℓ2 and X satisfies the following condition:
(Zτ) Every closed set AinX with w(A)< τ is aZ-set.
Proof. Since an ℓ2(Γ)-manifold has the τ-discrete cells property and ℓ2 × ℓ2(Γ) ≈ ℓ2(Γ), the “only if” part follows from Proposition 3.2.9 and the Stability Theorem 2.3.7.
To see the “if” part, letd∈Metr(X). We verify the condition in Lemma 3.5.8. To this end, let f :In×Γ →X be a map andε >0. For eachγ∈Γ, letfγ :In→X be the map defined byfγ(x) =f(x, γ). We may assume that Γ = (Γ,6) is a well-ordered set such that
card{
γ′∈Γ γ′ < γ}
< w(X) for everyγ∈Γ.
By the transfinite induction, we can define a map gγ : In → X so that d(gγ, fγ)< εand
δ(γ) = dist(
gγ(In),∪
γ′<γgγ′(In))
>0.
Indeed, putgγ0 =fγ0 forγ0= minΓ. Assume thatgγ′ has been defined for allγ′< γto satisfy the above condition. SinceA= cl∪
γ′<γgγ′(In) is aZ-set in X by (Zτ), we have a mapgγ :In→X\Asuch thatd(gγ, fγ)< ε. Then, f is ε-close to a map g : In×Γ →X defined by g(z, γ) = gγ(z). For each i ∈ N, let Γi ={γ ∈ Γ | δ(γ)> 2−i}. Then, (gγ(In))γ∈Γi is discrete in X.
SinceΓ =∪
i∈NΓi, it follows that (g(In× {γ}))γ∈Γ isσ-discrete inX. Thus, X satisfies the condition of Lemma 3.5.8, henceX is anℓ2(Γ)-manifold. ⊓⊔
The Kadec–Anderson Theorem 3.5.4 can be extended to the non-separable case as follows:
Theorem 3.5.13 (Kadec–Anderson–Toru´nczyk) Every infinite-dimen-sional Fr´echet space is homeomorphic to a Hilbert space with the same weight.
186 3 Hilbert Manifolds and Hilbert Cube Manifolds
Proof. Since the separable case has been proved, we prove the non-separable case. LetX be an infinite-dimensional non-separable Fr´echet space. We may assume thatX has a bounded invariant metricd. First, observe thatX con-tains an infinite-dimensional separable closed linear subspaceX0. Indeed, such anX0can be obtained as the closure of the linear span of a linearly indepen-dent countable subsetA⊂X. Note thatX0 is a separable Fr´echet space. By the Kadec–Anderson Theorem 3.5.4,X0≈ℓ2. By the Bartle–Graves–Michael Theorem 1.4.7,X ≈X1×ℓ2 for someX1, hence
X×ℓ2≈X1×ℓ2×ℓ2≈X1×ℓ2≈X.
To verify (Zτ), letAbe a closed set inXwithw(A)< w(X),f :In→Xbe a map andε >0. Sincew(f(In)) =ℵ0< w(X), it follows thatw(A−f(In))<
w(X), whence
A−f(In) ={
x−f(z)x∈A, z∈In}
is nowhere dense in X. Thus, we have y∈B(0, ε)\(A−f(In)). We define a map g :In →X \A byg(z) =f(z) +y for each z ∈In. Then,d(f, g)< ε.
Therefore,Ais aZ-set inX. It follows from Proposition 3.5.12 thatX is an ℓ2(Γ)-manifold, where cardΓ =w(X). SinceX is contractible,X≈ℓ2(Γ) by the Classification Theorem 2.6.1. ⊓⊔
Lemma 3.5.1 can be modified as follows:
Lemma 3.5.14 LetX = (X, d)be a path-connected metric space with a tower X1⊂X2⊂ · · · ⊂X satisfying the following properties:
(a) Given a compactum A ⊂ In, a map f : In → X with f(A) ⊂ Xi, and ε >0, there is a map g :In →Xj for some j >i such that f|A=g|A andd(f, g)< ε;
(b) Given ε > 0, there is some δ > 0 such that any map f : In → Xi
is ε-homotopic to a map g : In → Xj for some j > i such that dist(f(In), g(In))> δ.
Then, X has the discrete cells tower property.