152 3 Hilbert Manifolds and Hilbert Cube Manifolds
3.2 The Discrete (Disjoint) Cells Property 153 Proposition 3.2.1 For each n∈ω, the τ-locally finite n-cells property im-plies theτ-discreten-cells property.
Proof. We may assume thatΓ = (Γ,6) is a well-ordered set such that card{
γ′∈Γ γ′ < γ}
<cardΓ for everyγ∈Γ .
Let f : In ×Γ → X be a map and U ∈ cov(X). Since cardΓ > ℵ0, we have card(Γ ×Γ) = cardΓ. Applying the τ-locally finite cells property to the map ˜f :In×Γ ×Γ →X defined by ˜f(z, γ, γ′) =f(z, γ), we can obtain a map ˜g such that ˜g isU-close to ˜f and (˜g(In× {(γ, γ′)})(γ,γ′)∈Γ2 is locally finite in X. Note that each ˜g(In× {(γ, γ′)}) meets only finitely many other
˜
g(In× {(δ, δ′)}). By the transfinite induction, we can chooseδ(γ)∈Γ for each γ∈Γ so that
˜
g(In× {(γ, δ(γ))})∩˜g(In× {(γ′, δ(γ′))}) =∅ for γ′< γ.
Then, (˜g(In× {(γ, δ(γ))}))γ∈Γ is discrete inX because it is locally finite and mutually disjoint. Hence, the desired map g:In×Γ →X can be defined by g(z, γ) = ˜g(z, γ, δ(γ)). ⊓⊔
It is said that X has the disjoint n-cells property if every two maps f, g : In → X can be approximated by maps f′, g′ : In → X such that f′(In)∩g′(In) =∅. Namely, the disjointn-cells property can be defined by replacing Γ with2={0,1}. If X has the disjointn-cells property for every n∈N, it is said thatX has thedisjoint cells property.
Proposition 3.2.2 LetXbe a paracompact space andn, n1, . . . , nk ∈N(k>
2) with n1, . . . , nk 6n. IfX has the disjoint n-cells property, then all maps fi :Ini →X, i= 1, . . . , k, can be approximated by maps fi′ :Ini →X such that fi′(Ini)∩fj′(Inj) =∅ fori̸=j.
Proof. We shall prove the proposition by induction onk. To see the casek= 2, form < n, letpm:In→Im andim:Im→In be the maps defined by
pm(x) = (x(1), . . . , x(m)) and im(x) = (x(1), . . . , x(m),0, . . . ,0).
We writepn =in= idIn for convenience. Letfj:Inj →X,j = 1,2, be maps and U ∈cov(X). By the disjoint n-cells property, the maps fjpnj :In →X areU-close to mapsgj :In→Xsuch thatg1(In)∩g2(In) =∅. Then, the maps fj′ =gjinj :Inj →X areU-close tofj=fjpnjinj and f1′(In)∩f2′(In) =∅.
Now, assuming the case k−1, we prove the case k. Let fi : Ini → X, i = 1, . . . , k, be maps. For each U ∈ cov(X), take its star-refinement V ∈ cov(X). By the inductive assumption, the maps f1, . . . , fk−1 are V-close to maps fi′ : Ini → X, i = 1, . . . , k−1, such that fi′(Ini)∩fj′(Inj) = ∅ for 16i < j < k. TakeV′ ∈cov(X) so that
V′≺ V ∧{
X\f1′(In1), X\∪k−1
i=2 fi′(Ini)} .
154 3 Hilbert Manifolds and Hilbert Cube Manifolds
Using the inductive assumption again, the maps f2′, . . . , fk−1′ , fk areV′-close to mapsfi′′ :Ini →X, i= 2, . . . , k, such that fi′′(Ini)∩fj′′(Inj) =∅ for 1<
i < j6k. Then, observe thatf1′(In1)∩fi′′(In) =∅for everyi= 2, . . . , k−1.
TakeV′′∈cov(X) so that V′′≺ V ∧{
X\(f1′(In1)∪fk′′(Ink)), X\∪k−1
i=2 fi′′(Ini)} .
From the casek= 2, the mapsf1′ andfk′′areV′′-close to maps f1′′′ :In1 →X andfk′′′:Ink →X such thatf1′′′(In1)∩fk′′′(Ink) =∅. Observe thatf1′′′(In1)∩ fi′′(Ini) = ∅ and fk′′′(In1)∩fi′′(Ini) =∅ for every i = 2, . . . , k−1. For each 1 < i < k, let fi′′′ = fi′′. Then, it follows that fi′′′(In)∩fj′′′(In) = ∅ for 16i < j6k. Moreover, everyfi′′′ isU-close tofi because stV ≺ U. ⊓⊔
SinceIn×Γ ∈M1(τ) andIn×{0,1} ∈M0, theM1(τ)-universality implies theτ-discrete cells property, and theM0-universality implies the disjoint cells property.
Remark 3.1 Let E be an infinite-dimensional normed linear space with densE > τ. By Proposition 2.1.6, we can take Γ as a δ-discrete set in the sphere SE of E, where 0 < δ < 1. Taking an embedding h : In → E with h(In) ⊂ (δ/3)BE, we can define a closed embedding f : In ×Γ → E by f(x, γ) =h(x) +γ. Thus, eachIn×Γ can be embedded inE as a closed set, that is,In×Γ ∈ FE. Consequently, theFE-universality implies theτ-discrete cells property.
For a simplexσwith dimσ >0 andt∈I, the following notation is useful:
σ[t] ={
(1−s)ˆσ+syy∈∂σ, 06s6t} . Then,σ[0] ={ˆσ},σ[1] =σ, andσ[t]⊂rintσfor every t <1.
Lemma 3.2.3 LetX be an ANR with the τ-discrete cells property andK be a finite-dimensional simplicial complex with cardK(0) 6τ. Suppose that (1) X isQ-stable (i.e.,X×Q≈X) or(2)all maps from In,n∈N, toX can be approximated by maps with strongZ-set images. Then, each mapf :|K| →X can be approximated by OK-maps.
Proof. By induction on dimK, we shall show the proposition. It is easy to see that theτ-discrete 0-cells property implies the case dimK= 0. Assume that the proposition is valid for any (n−1)-dimensional simplicial complex. Let dimK=nandf :|K| →X be a map. For eachU ∈cov(X), letV ∈cov(X) be a star-refinement ofU.
First, by the inductive assumption and the Homotopy Extension Theorem 1.8.10, we have a map f′ : |K| → X such that f′||K(n−1)| is an OK(n−1) -map andf′ ≃V f. Since OK(n−1) =OK|K(n−1), we apply Proposition 3.1.11 to obtain an open neighborhood R of |K(n−1)| in |K| such that f′|R is an (OK|R)-map, that is,f′ is belongs to the following subset of C(|K|, X):
3.2 The Discrete (Disjoint) Cells Property 155 {g∈C(|K|, X)g|Ris an (OK|R)-map}
.
This set is open in C(|K|, X) because COK|R(R, X) is open in C(R, X) by Lemma 3.1.7 and the restriction operator of C(|K|, X) to C(R, X) is continu-ous by Proposition 3.1.3(2). Hence, we can findV′∈cov(X) such thatV′≺ V and if a mapg′:|K| →X isV′-close tof′, theng′|Ris also an (OK|R)-map.
In case (2), replacingV′ by its star-refinement, we can require the condition ofV′ such that if a mapg′:|K| →X is stV′-close tof′, theng′|R is also an (OK|R)-map.
Next, using theτ-discreten-cells property, we can obtain maps fσ :σ→ X,σ∈K\K(n−1), such that {fσ(σ)|σ∈K\K(n−1)} is discrete inX and fσ ≃V′ f′|σ, where we add in case (2) the condition that fσ(σ) is a strong Z-set in X. For each σ∈K\K(n−1), let hσ :σ×I→X be a V′-homotopy with hσ0 =f′|σandhσ1 =fσ. For each σ∈K\K(n−1), choose tσ ∈(0,1) so that cl(σ\σ[tσ]) ⊂R and define a map gσ : σ→ X by gσ|σ[tσ] = fσ|σ[tσ] and
gσ((1−s)ˆσ+sy) =hσ (
(1−s)ˆσ+sy, 1−s 1−tσ
)
for (y, s)∈∂σ×[tσ,1].
Then, {gσ(σ[tσ]) | σ ∈ K\K(n−1)} is discrete in X, and in case (2), each gσ(σ[tσ]) is a strongZ-set inX. Observe thatgσ|∂σ=f′|∂σ andgσ≃V′ f′|σ rel.∂σ.
In case (1), take a Urysohn mapγ:|K| →Isuch that γ( ∪
σ∈K\K(n−1)cl(σ\σ[tσ]))
= 0 and γ(|K| \R) = 1.
We can define a map ˜g:|K| →X×Ias follows:
˜ g(x) =
{(f′(x),0) ifx∈ |K(n−1)|, (gσ(x), γ(x)) ifx∈σ∈K\K(n−1).
Then, prXg˜≃V′ f′ because prXg|σ˜ =gσ ≃V′ f′|σrel. ∂σ. Hence, prX˜g|R is an (OK|R)-map. It should be remarked that ˜g−1(X×[0,1))⊂Rand
˜
g−1(X×(0,1])⊂ |K| \ ∪
σ∈K\K(n−1)
cl(σ\σ[tσ])⊂ ∪
σ∈K\K(n−1)
σ[tσ].
To see that ˜g is an OK-map, let (x, t) ∈ X ×I. Then, x has an open neighborhoodW in X such that
˜
g−1(W×I)∩R= (prX˜g|R)−1(W)⊂OK(v) for somev∈K(0) andW meets at most one member of
156 3 Hilbert Manifolds and Hilbert Cube Manifolds
{gσ(σ[tσ])|σ∈K\K(n−1)}={prXg(σ[t˜ σ])|σ∈K\K(n−1)}, where the both cases ˜g−1(W×I)∩R=∅andW∩∪
σ∈K\K(n−1)gσ(σ[tσ]) =∅ are possible. Whent <1,W×[0,1) is an open neighborhood of (x, t) inX×I.
Since ˜g−1(W×[0,1))⊂R, it follows that
˜
g−1(W×[0,1))⊂g˜−1(W×I)∩R⊂OK(v).
Whent >0,W×(0,1] is an open neighborhood of (x, t) inX×I. Observe
˜
g−1(W ×(0,1])⊂˜g−1(pr−1X (W)) = (prXg)˜ −1(W), SinceW meets at most one prX˜g(σ[tσ]) and
˜
g−1(W×(0,1])⊂ ∪
σ∈K\K(n−1)
σ[tσ],
˜
g−1(W×(0,1]) is contained one prX˜g(σ[tσ]), hence ˜g−1(W×(0,1])⊂OK(v) forv∈σ(0).
Since X is Q-stable, the projection prX : X ×I → X is V′-close to a homeomorphism h: X×Iby Corollary 2.3.11. Then, g =h˜g : |K| →X is also anOK-map. Moreover,g isV′-close to prX˜g, prX˜g isV′-close tof′, and f′ isV-close tof. SinceV′ ≺ Vand stV ≺ U, it follows thatgisU-close tof. In case (2), let g′ = prXg˜ : |K| → X, that is, g′|σ = gσ for each σ ∈ K\K(n−1). Then, g′ ≃V′ f′ and{g′(σ[tσ])| σ∈K\K(n−1)} is discrete in X. Since eachg′(σ[tσ]) =fσ(σ[tσ]) is a strong Z-set inX, the discrete union Z =∪
σ∈K\K(n−1)g′(σ[tσ]) is also a strongZ-set inXby Corollary 2.8.15. Due to Proposition 2.8.12, there is a homotopyφ:X×I→X such thatφ0= id, φ(X×(0,1])∩Z=∅, andφis closed overZ (i.e.,Z∩clφ(A) =Z∩φ(A) for any closed setA in X×I). By Lemma 2.5.2, we have a map α:X →(0,1]
such that {
{x} ×[0, α(x)]x∈X}
≺φ−1(V′).
Since {g′(σ[tσ]) | σ ∈ K\K(n−1)} is discrete in X, X has a discrete open collection{Wσ|σ∈K\K(n−1)}such thatg′(σ[tσ])⊂Wσ for everyσ∈K\ K(n−1). Then,σ[tσ]⊂g′−1(Wσ). Since|K|is perfectly normal by Proposition 1.5.2, there is a mapβ:|K| →Isuch that
β−1(0) = ∪
σ∈K\K(n−1)
σ[tσ] and β−1(1) =|K| \ ∪
σ∈K\K(n−1)
(rintσ∩g′−1(Wσ)) ,
Then, observe that
3.2 The Discrete (Disjoint) Cells Property 157 β−1((0,1]) =|K| \ ∪
σ∈K\K(n−1)
σ[tσ] = ∪
σ∈K\K(n−1)
(σ\σ[tσ])
= ∪
σ∈K\K(n−1)
kσ(∂σ×[0,1))⊂R and β−1([0,1)) = ∪
σ∈K\K(n−1)
(rintσ∩g′−1(Wσ)) . We now define a mapg:|K| →X by
g(x) =φ(g′(x), αg′(x)·β(x)) for eachx∈ |K|.
Then,g|σ[tσ] =g′|σ[tσ] for eachσ∈K\K(n−1), soZ =∪
σ∈K\K(n−1)g(σ[tσ]).
Moreover,g isV-close tof becauseg isV′-close tog′, g′ isV′-close to f′,f′ is V-close to f, V′ ≺ V, and stV ≺ U. Since g is stV′-close tof′, it follows that g|R is also anOK-map.
It remains to show thatg is anOK-map. Eachx∈X has an open neigh-borhood W in X such that (g|R)−1(W) is contained in OK(v) for some v ∈ K(0) (the case (g|R)−1(W) = ∅ is possible). When x ∈ X \ Z = X \∪
σ∈K\K(n−1)g(σ[tσ]), W \Z = W ∩(X \Z) is an open neighborhood ofxinX and
g−1(W\Z) =g−1(W)\ ∪
σ∈K\K(n−1)
g−1(g(σ[tσ]))
⊂g−1(W)\ ∪
σ∈K\K(n−1)
σ[tσ]
=g−1(W)∩ (
|K| \ ∪
σ∈K\K(n−1)
σ[tσ] )
=g−1(W)∩ ∪
σ∈K\K(n−1)
(σ\σ[tσ])
⊂g−1(W)∩R= (g|R)−1(W)⊂OK(v).
Whenx∈Z,xis contained in only oneg(σ[tσ]). Then,Wσis an open neigh-borhood of x in X. Note that φ(x,0) = x and x ̸∈ φ(X ×(0,1]) because x∈Z. Hence,φ−1(x) ={(x,0)}. Consider the following closed set inX×I:
A={(x′, t)∈Wσ×I|t>α(x′)}.
Then, Z∩φ(A) =∅. Sinceφ :X ×I→X is closed overZ, it follows that Z ∩clφ(A) = ∅. Hence, x has an open neighborhood W0 in X such that W0⊂Wσ andW0∩φ(A) =∅, so
φ−1(W0)⊂{
(x′, t)∈Wσ×It < α(x′)} .
For each y ∈g−1(W0), (g′(y), αg′(y)β(y))∈ φ−1(W0), which means β(y) <
1. Since g′(y) ∈ W0 ⊂ Wσ, it follows that y ∈ rintσ∩g′−1(Wσ). Hence, g−1(W0)⊂rintσ⊂OK(v) forv∈σ(0). Thus,g is anOK-map. ⊓⊔
158 3 Hilbert Manifolds and Hilbert Cube Manifolds
Note that the disjoint cells property means them-discrete cells property for everym∈N(cf. Proposition 3.2.2). Hence, the proof above is available, even if the τ-discrete cells property and the condition cardK(0) 6τ are replaced by the disjoint cells property and the condition cardK(0) <ℵ0 (i.e.,K is a finite simplicial complex). Thus, we have the following:
Lemma 3.2.4 LetX be a separable completely metrizable ANR with the dis-joint cells property, andK be a finite simplicial complex. Suppose that(1) X is Q-stable (i.e.,X×Q≈X) or(2) all maps from In,n∈N, to X can be approximated by maps with strong Z-set images. Then, for any U ∈cov(X), each map f :|K| →X isU-close to anOK-map. ⊓⊔
Due to the following proposition, a separable completely metrizable ANR X has the disjoint cells property if and only if all maps fromIn, n ∈N, to X can be approximated by maps withZ-set images. This condition is weaker than condition (2) of Lemma 3.2.4 above.
Proposition 3.2.5 For each separable completely metrizable ANR X, the following are equivalent:
(a) X has the disjoint cells property;
(b) Every maps f, g : Q → X are arbitrarily close to maps f′, g′ : Q → X withf′(Q)∩g′(Q) =∅;
(c) C(Q, X) has no isolated points and C(Q, X) contains a countable dense set{fi|i∈N} such that fi(Q)∩fj(Q) =∅ if i̸=j;
(d) Every mapf :Q→X can be approximated by mapsg:Q→X such that g(Q)is aZ-set inX;
(e) For each n ∈ N, every map f : In → X can be approximated by maps g:In→X such that g(In)is aZ-set inX;
(f) For each n ∈ N, every map f : In → X can be approximated by maps g:In→X such that g(In)is aZn-set inX.
Proof. The implications (e)⇒(f)⇒(a) are trivial. In condition (c),{fi|i∈ N} \ {fj} is dense in C(Q, X) for each j ∈ N because fj is not isolated in C(Q, X). Then, it follows from Theorem 2.8.6 thatfj(Q) is aZ-set inX for eachj∈N. So, the implication (c)⇒(d) holds.
For eachn∈N, letpn :Q→In andin:In→Qbe the maps defined by pn(x) = (x(1), . . . , x(n)) and in(x) = (x(1), . . . , x(n),0,0, . . .).
Then,pnin= id, from which the implication (d)⇒(e) can be easily obtained.
On the other hand,inpn converges to idQ asn→ ∞. Using this fact, we can easily see the implication (a)⇒(b).
(b)⇒(c): We may assume thatX = (X, d) is a separable complete metric space. The space C(Q, X) with the sup-metric is complete, where the same letterdstands for the sup-metric. It follows easily from (b) that C(Q, X) has
3.2 The Discrete (Disjoint) Cells Property 159 no isolated points. Since C(Q, X) is separable metrizable, it has a countable dense set {gi |i ∈N}. Then, {gi |i > k} is also dense in C(Q, X) for each k∈N. We define
M ={
(fi)i∈N∈C(Q, X)Nd(fi, gi)<2−i for eachi∈N} .
Since M is a Gδ-set in the product space C(Q, X)N, M is also completely metrizable. For eachj < k, we have the following open set:
Uj,k={(fi)i∈N∈M |fj(Q)∩fk(Q) =∅} ,
which is dense inM by (b). By the Baire Category Theorem 1.3.11,∩
j<kUj,k
is dense in M. In particular, we have (fi)i∈N ∈ ∩
j<kUj,k. Then, the set {fi|i∈N}satisfies condition (c). ⊓⊔
Now, we can prove that the disjoint cells property characterizes theM0 -universality for separable completely metrizable ANRs, that is,
Proposition 3.2.6 Let X be a separable completely metrizable ANR such that everyZ-set inX is a strongZ-set. Then,X is M0-universal if and only if X has the disjoint cells property.
Proof. SinceIn×{0,1} ∈M0, the “only if” part is trivial. IfXhas the disjoint cells property, then all maps from In, n∈N, to X can be approximated by maps with Z-set images by Proposition 3.2.5. Because we assume that every Z-set in X is a strong Z-set, the “if” part follows from Lemma 3.2.4 and Theorem 3.1.14. ⊓⊔
As is well known, every component of a locally connected space is clopen.
By an analogous proof, we can easily show that every locally compact metriz-able space is a discrete union of locally compact and σ-compact metrizable spaces, where σ-compact metrizable spaces are separable because compact metrizable spaces are separable.
Indeed, a locally compact metrizable spaceX has a locally finite open cover U such that clU is compact for every U ∈ U. Then, note that U is star-finite (i.e., U[U] is finite for every U ∈ U). For each U ∈ U, we define U∗ = ∪
n∈Nstn(U,U), where each stn(U,U) = st(stn−1(U,U),U), st0(U,U) = U (hence st1(U,U) = st(U,U)). Then, U∗ is an equivalence class with respect to the equivalence relation∼onX defined as follows:
x∼y def⇔ ∃U1, . . . , Un∈ U such that
x∈U1, y∈Un, Ui∩Ui+1̸=∅, i= 1, . . . , n−1.
Thus, for eachU, V ∈ U,U∗=V∗⇔U∗∩V∗̸=∅. Hence,U∗is clopen inX for eachU ∈ U. SinceUis star-finite and clU is compact for everyU ∈ U, it follows that cl stn(U,U),n∈N, are compact. Then,U∗=∪
n∈Ncl stn(U,U) because cl stn−1(U,U)⊂stn(U,U). Therefore,U∗isσ-compact. SinceU∗is open inX,U∗is locally compact.
160 3 Hilbert Manifolds and Hilbert Cube Manifolds
WhenX is a locally compact ANR, every Z-set in X is a strong Z-set (Proposition 2.8.1) and each component ofX is clopen and separable as saw in the above. Thus, the following characterization of the M0-universality is the direct consequence of Proposition 3.2.6:
Theorem 3.2.7 A locally compact ANRX is M0-universal if and only ifX has the disjoint cells property. ⊓⊔
Furthermore, we can prove the following:
Proposition 3.2.8 Let X be a locally compact ANR with the disjoint cells property and Y be a locally compact metrizable space. Then, for each U ∈ cov(X), each proper map f :Y →X isU-close to a (strong)Z-embedding.
Proof. As saw in the above, each component ofX is clopen andσ-compact.
Sincef is proper, the inverse image of each component ofX isσ-compact, so separable. By restricting f on the inverse image of each component of X, it can be assumed thatX andY are separable.
By virtue of Proposition 3.1.13 and Theorem 3.1.15, it suffices to show that COK(|K|, X) is dense in CP(|K|, X) for any countable locally finite simplicial complex K, where COK(|K|, X) ⊂ CP(|K|, X) because K is locally finite.
Since the case whereK is finite has already been obtained by Theorem 3.2.7, we may consider the case whereKis infinite, i.e., cardK=ℵ0. Then, we can writeK={σi|i∈N}. For eachn∈N, we define
An ={
σi[1−2−n]i6n} ,
where the notation σ[t] was defined before Proposition 3.2.3 and we regard σi[1−2−n] =σi if dimσi= 0. Then,An is a pairwise disjoint finite collection of cells in|K|. Note that the space CP(|K|, X) of all proper maps is open in C(|K|, X) (Proposition 3.1.5). SinceX has the disjoint cells property and is an ANR, we can apply Proposition 3.2.2 and the Homotopy Extension Theorem 1.8.10 to see that the following set is dense in CP(|K|, X):
Fn={
f ∈CP(|K|, X)f(An) is a pairwise disjoint collection} . It should be observed that Fn is open in CP(|K|, X). Since CP(|K|, X) is a Baire by Theorem 3.1.6, ∩
n∈NFn is dense in CP(|K|, X).
For each g ∈ ∩
n∈NFn and x ∈ g(|K|), there is a unique σi such that g−1(x)⊂rintσi. Indeed,x∈g(rintσi)∩g(rintσj) for some i̸=j. Then,
x∈g(σi[1−2−n])∩g(σj[1−2−n]) for some n>i, j.
This contradicts the fact that g∈Fn (i.e., g(An) is pairwise disjoint). Thus, we have {g−1(x) | x ∈ g(|K|)} ≺ OK, hence g is an OK-map because g is a closed map (Proposition 1.3.1). Therefore, ∩
n∈NFn ⊂COK(|K|, X), hence COK(|K|, X) is dense in CP(|K|, X). ⊓⊔