112 2 Fundamental Results on Infinite-Dimensional Manifolds
· · ·
X
2−1 1 3−1 0
n−1 A
· · · ·
(2n)−1 A
Fig. 2.18.AZ-set being not a strongZ-set
above,A is called astrong Z-set in X. A countable union ofZ-sets (resp.
strongZ-sets) is called aZσ-set(resp.strongZσ-set). Every closed subset of a (strong) Z-set is also a (strong) Z-set and every Fσ-set contained in a (strong)Zσ-set is a (strong)Zσ-set. Obviously a strongZ-set is aZ-set. The converse is also true in a locally compact paracompact space, that is, Proposition 2.8.1 EveryZ-set in a locally compact paracompact spaceX is a strongZ-set. In particular, everyZ-set in aQ-manifold is a strongZ-set.
Proof. Each U ∈ cov(X) has a locally finite open refinement V ∈ cov(X) such that clV is compact for each V ∈ V. Then, idX is V-close to a map f : X →X \A. Since idX is a proper map,f is also proper by Proposition 2.1.17, hence it is perfect by Proposition 1.3.1. Thus, f(X) is closed in X, henceA∩clf(X) =A∩f(X) =∅. ⊓⊔
In general, a (compact) Z-set is not a strong Z-set even in an AR. For example, let
X=I× {0} ∪{
n−1n∈N}
×I⊂R2.
Then, X is an AR and A={(0,0)}is aZ-set, but not a strongZ-set in X.
— Fig. 2.18.
Indeed, for eachn∈N, we have a mapfn:X →X\Adefined as follows:
fn(s, t) =
(s, t) ifs>n−1 ort>n−1,
(s,2t−n−1) ifs < n−1 and (2n)−16t6n−1, (2nts+ (1−2nt)n−1,0) ifs < n−1 and 0< t6(2n)−1, (n−1,0) ifs6n−1 andt= 0,
wheret >0 impliess=m−1 for somem∈N. Then,fn is√
2n−1-close to idX with respect to the Euclidean metric. Hence,Ais aZ-set inX. On the other hand, assume that a mapf:X→X isU-close to idX, where
U={
{n−1} ×(2−1,1]n∈N}
∪{
X∩[0,1)2, (2−1,1]×[0,1)}
∈cov(X).
2.8 Z-Sets and StrongZ-Sets in ANRs 113 For eachn∈N, letvn= (n−1,1)∈X. Sincef(vn)∈ {n−1} ×(2−1,1] and f(v1)∈(2−1,1]×[0,1), the imagef(X) contains every point (n−1,0)∈X, which implies (0,0)∈clXf(X). Thus,Ais not a strongZ-set inX.
Lemma 2.8.2 LetAbe a closed set in a metrizable spaceX. IfAis contained in some collared setC in X, then Ais a strongZ-set.
Proof. Let k :C×[0,1) →X be a collar of C in X. Then, A has an open neighborhood W in X with clW ⊂ k(C×[0,1)). For each U ∈ cov(X), choose a map γ : C → I so that γ(x) > 0 for x ∈ A, γ(x) = 0 for x ∈ C\W, and{{x} ×[0, γ(x)]|x∈C∩W} ≺ k−1(U|W) (Lemma 2.5.2). Let V = {k(x, t) | t < γ(x)} ⊂ W. Observe that V is open in X, A ⊂ V, and clV = {k(x, t) | t 6 γ(x)}. We can define a retraction f : X → X \V as follows: f|X \V = id and f(k(x, t)) =k(x, γ(x)) if t6γ(x). Then, f isU -close to id andf(X) =X\V is a closed set inX that missesA. Hence,Ais a strongZ-set inX. ⊓⊔
In a metrizable space, every collared set is anFσ-set because it is locally closed, that is, closed in an open set. Thus, by Lemma 2.8.2 above, we have the following:
Proposition 2.8.3 Every collared set in a metrizable space is a strong Zσ -set. ⊓⊔
The following is a combination of Theorem 2.4.9 and Lemma 2.8.2:
Proposition 2.8.4 EveryE-deficient closed (orFσ-) set in anE-stable space is a strongZ-set (orZσ-set). ⊓⊔
Let n ∈ N. A closed set A in a space X is called a Zn-set if, for each U ∈ cov(X), every map f : In →X is U-close to a map f′ : In → X\A.
For everyk6n∈N, eachZn-set inX is aZk-set in X. Indeed, in the case k < n, letp= prIk :In =Ik×In−k →Ik be the projection and i:Ik →In be the embedding defined by i(x) = (x,0) ∈Ik×In−k for each x∈ Ik. For each mapf :Ik →X and U ∈cov(X), we have a mapf′ :In →X\A that isU-close tof p. Then, f′i:Ik →X\Ais U-close tof =f pi.
Ik
i
f //X
In
p
FF
f_′ _//
_ X\A
∪
WhenAis aZn-set inX for everyn∈N, we callAa Z∞-setin X.
Proposition 2.8.5 Every locally finite union ofZn-sets (resp.Z∞-sets) in a paracompact spaceX is also a Zn-set (resp. a Z∞-set).
114 2 Fundamental Results on Infinite-Dimensional Manifolds
Proof. First of all, observe that if theZn-set case holds for everyn∈N, then theZ∞-set case follows from the definition.
By induction, the finite case reduces to to showing that if A and B are Zn-sets in X then A∪B is also a Zn-set in X. For each U ∈ cov(X), let V ∈ cov(X) be a star-refinement of U. Each map f : In →X is V-close to a map f′ :In →X\A. Let W ∈cov(X) be a common refinement ofV and {X\A, X\f′(In)}. Then, f′ is W-close to a mapf′′ : In →X\B. Since f′(In)⊂X\A, it follows that f′′(In)⊂X\A. Note thatf′′isU-close tof. Thus,A∪B is a Zn-set inX.
We now show the locally finite case. LetA=∪
λ∈ΛAλ be a locally finite union of Zn-sets Aλ in X andU ∈cov(X). For each mapf :In →X, since f(In) is compact,f(In) has an open neighborhoodU that meets only finitely manyAλ, that is, there areλ1, . . . , λn∈Λ such that
U ∩A=U∩(Aλ1∪ · · · ∪Aλn).
Let V ∈ cov(X) be a common refinement of U and {U, X \f(In)}. Since Aλ1∪ · · · ∪Aλn is aZn-set inX,f isV-close to a map
f′:In→X\(Aλ1∪ · · · ∪Aλn).
Then,f′(In)⊂X\Abecause f′(In)⊂U. Hence,Ais aZn-set inX. ⊓⊔ In an ANR,Z-sets are characterized as follows:
Theorem 2.8.6 For a closed setA in an ANRX, the following are equiva-lent:
(a) Ais aZ-set inX;
(b) For eachε >0, each map f :Q→X is ε-close to a mapg:Q→X\A with respect to any given metricd∈Metr(X);
(c) Ais aZ∞-set inX;
(d) For an arbitrary simplicial complex K and any U ∈ cov(X), each map f :|K| →X isU-close to a map g:|K| →X\A;
(e) For a finite simplicial complex K and any U ∈ cov(X), each map f :
|K| →X isU-close to a mapg:|K| →X\A;
(f) The inclusionX\A⊂X is a hereditary weak homotopy equivalence, i.e., for every open set U in X, the inclusion U\A⊂U is a weak homotopy equivalence;
(g) For every open set U in X, the inclusion U \ A ⊂ U is a homotopy equivalence;
(h) X\A is homotopy dense inX;
(i) X\Ais dense inX and, for each a∈A, each neighborhoodU ofa inX contains a neighborhoodV in X such that every map α:Sn−1→V \A is null-homotopic inU\A for everyn∈N.
2.8 Z-Sets and StrongZ-Sets in ANRs 115 Proof. The implications (a) ⇒ (b) ⇒ (c) and (d) ⇒ (e) ⇒ (c) are trivial.
The equivalence (f) ⇔ (g) follows from the fact that every weak homotopy equivalence between ANRs is a homotopy equivalence (Corollary 1.8.19). The equivalences (f) ⇔ (h) ⇔(i) are direct consequences of Corollary 1.9.4. By Hanner’s Characterization of ANRs (Theorem 1.8.15), for eachU ∈cov(X), we have a simplicial complex K with maps f : X → |K| and g : |K| → X such that gf isU-close to idX. Then, it is easy to see the implication (d)⇒ (a). Thus, it remains to show the implication (c)⇒(d) and the equivalence (e)⇔(f).
(a)
triv.
(1.8.15)(d)
ks triv. +3(e)
triv.
z}}}}}}}
}}}}}}}
ks +3(f)
KS
(1.8.19)
ks(1.9.4)+3(h)ks(1.9.4)+3(i) (b) triv. +3(c)
KS
(g)
(c) ⇒ (d): For each U ∈ cov(X), we adopt a metric d ∈ Metr(X) such that{Bd(x,1)|x∈X} ≺ U (1.3.18). Then, it suffices to show that each map f : |K| →X is 1-close to a map g :|K| →X \A. We shall construct maps gn :|K| →X,n∈ω, such that
gn(|K(n)|)⊂X\A, gn||K(n−1)|=gn−1||K(n−1)|
and gn is 2−n−1-close to gn−1, where g−1 =f. Then, (gn)n∈N is uniformly convergent to the map g:|K| →X defined byg||K(n)|=gn||K(n)|for each n∈ω, whence gis 1-close tof and
g(|K|) = ∪
n∈ω
gn(|K(n)|)⊂X\A.
Assume that g−1, g0, . . . , gn−1 have been obtained. By Corollary 1.8.11, we have V ∈ cov(X) that is an h-refinement of an open cover of X with mesh <2−n−1. Then, any two V-close maps are 2−n−1-homotopic. For each n-simplex σ∈K, chooseε >0 so that{B(gn−1(x), ε)|x∈σ} ≺ Vand let
δ= min{
ε/2, dist(gn−1(∂σ), A)}
>0.
Since X is an ANR, we apply (c) to obtain a map gσ : σ → X \A with gσ ≃δ gn−1|σinX. It follows from the definition ofδthat gσ|∂σ≃δ gn−1|∂σ inX\A. SinceX\Ais also an ANR, we can apply the Homotopy Extension Theorem 1.8.10 to extend gn−1|∂σ to a mapgσ′ :σ→X\Awith gσ′ ≃δ gσ. Note that g′σ and gn−1|σare V-close becaused(g′σ(x), gn−1(x))<2δ6εfor eachx∈σ. Then, we have a mapgn′ :|K(n)| →X\Adefined bygn′||K(n−1)|= gn−1||K(n−1)| and g′n|σ = gσ′ for each n-simplex σ ∈ K, whence g′n is V-close to gn−1||K(n)|, which implies g′n ≃2−n−1 gn−1||K(n)| in X. Applying the Homotopy Extension Theorem 1.8.10 again, we can extendg′n to a map gn :|K| →X that is 2−n−1-close togn−1. This completes the induction.
116 2 Fundamental Results on Infinite-Dimensional Manifolds
(e)⇒(f): For each open setU in X andi∈N, letβ :Bi →U be a map with α=β|Si−1:Si−1 →U \A. SinceU is an ANR and β(Bi) is compact, we can chooseε >0 so that a map ofBi toU extendingαis homotopic toβ rel.Si−1 whenever it isε-close toβ (Theorem 1.8.11). Let
δ= min{
ε/2, dist(β(Bi), X\U), dist(α(Si−1), A)}
>0.
We apply (e) with Theorem 1.8.11 to obtain a map γ :Bi →X\A that is δ-homotopic to β in X, hence inU \A. Observe thatγ|Si−1 is δ-homotopic toαinU\A. SinceU\Ais an ANR, we can apply the Homotopy Extension Theorem 1.8.10 to extend α to a map ¯α : Bi →U \A that is δ-homotopic to γ. Since ¯αisε-close toβ, ¯α≃β rel.Si−1. This means that the inclusion U\A⊂U is a weak homotopy equivalence.
(f) ⇒(e): Let dimK =n. For eachU ∈cov(X), we take open covers of X as follows:
U =Un
≻ U∗ n−1
≻ · · ·∗ ≻ U∗ 1
≻ U∗ 0
≻ U∗ −1.
For each map f : |K| → X, by subdividing K, we can assume that K ≺ f−1(U−1), that is,f(K) ={f(σ)|σ∈K} ≺ U−1(Theorem 1.5.8).
Letf−1=f, and assume that we have a mapfi−1:|K| →X such that fi−1(|K(i−1)|)⊂X\A, fi−1(K)≺ Ui−1 and fi−1≃Ui−1 f.
For each i-simplex σ ∈K, we can chooseUσ ∈ Ui−1 so that fi−1(σ) ⊂Uσ, whencefi−1(∂σ)⊂Uσ\A. Since the inclusionUσ\A⊂Uσ is ani-homotopy equivalence, there is a map fσ : σ → Uσ\ A such that fσ|∂σ = fi−1|∂σ and fσ ≃ fi−1|σ rel. ∂σ in Uσ. We define a map fi′ : |K(i)| → X \A by fi′||K(i−1)|=fi−1||K(i−1)| andfi′|σ=fσ for eachi-simplex σ∈K. Observe thatfi′ ≃Ui−1 fi−1||K(i)|rel.|K(i−1)|inX. Then, by the Homotopy Extension Theorem 1.8.10, we can extend fi′ to a map fi : |K| → X that is Ui−1 -homotopic tofi−1, so Ui-homotopic tof. Moreover,
fi(K)≺st(fi−1(K),Ui−1)≺stUi−1≺ Ui.
By induction, we have a mapfn:|K|=|K(n)| →X\Athat isU-close to f. Thus, (e) holds. ⊓⊔
Remark 2.9 WhenX is anE-manifold, each neighborhoodU of any point ofX contains a homotopically trivial open neighborhoodV, whereV is said to be homotopically trivial if for each n ∈ ω, every map α: Sn → V is null-homotopic. Then, (i) is derived from the following condition, which is weaker than (f) (or (g)):
(z) IfU is a homotopically trivial open set inX, thenU\Ais also homotopi-cally trivial.
Thus, the above (z) characterizes Z-sets in an E-manifold, and this was the first definition due to R.D. Anderson.
2.8 Z-Sets and StrongZ-Sets in ANRs 117 Using the above characterization ofZ-sets (2.8.6), the following is easily shown:
Corollary 2.8.7 Let X be an ANR.
(i) A locally finite union ofZ-sets inX is aZ-set.
(ii) For eachZ-setAinX and each open setU inX,A∩U is aZ-set inU. (iii) A closed set A⊂X is aZ-set inX if each x∈A has a neighborhoodU
inX such thatA∩U is aZ-set inU. ⊓⊔
In the rest of this section, we shall show some properties of strongZ-sets.
Different to the Z-set case, the following can be directly derived from the definition:
Proposition 2.8.8 (1) A finite union of strong Z-sets in a paracompact spaceX is also a strong Z-set.
(2) A closed set A in a metrizable space X is a strong Z-set in X if A is a strongZ-set in an open neighborhoodU of Ain X.
Proof. (1): It suffices to show that A∪B is a strong Z-set in X for strong Z-sets A and B in X. For each open cover U ∈ cov(X), let V ∈ cov(X) be an open star-refinement of U. We have a map f : X → X and an open neighborhoodV ofAinX such thatf isV-close to id and clV∩clf(X) =∅.
ChooseW ∈cov(X) so thatW ≺ V andW ≺ {X\clf(X), X\clV}. Now, we have a mapg:X →Xsuch thatgisW-close to id andB∩clg(X) =∅. Then, gf isU-close to id andB∩clgf(X) =∅. Sincegis{X\clf(X), X\clV}-close tof, we havegf(X)⊂X\clV, henceA∩clgf(X) =∅. Thus, it follows that (A∪B)∩clgf(X) =∅.
(2): For each open coverU ∈cov(X), we can takeV ∈ cov(U) such that V ≺ U|U and V is fitting in X by Proposition 2.1.16. Since A is a strong Z-set in U, we have a map f : U → U and an open neighborhood W of A in U such that f is V-close to id and W ∩f(U) = ∅. Then, f can be extended to a map ˜f :X →X by ˜f|X\U = id, where ˜f isU-close to id and W ∩f˜(X) =W ∩U ∩f˜(X) =W ∩f(U) =∅. Therefore,Ais a strongZ-set in X. ⊓⊔
To see that strongZ-sets have the same properties as in Corollary 2.8.7 above, it is necessary to introduce a generalization of closed maps. A map f : X → Y is said to be closed over C ⊂ Y if C ∩f(A) = C∩clf(A) for every closed set A in X. Note thatC∩f(A) = C∩clf(A) means that C∩f(A) =∅ impliesC∩clf(A) =∅.
Lemma 2.8.9 A map f : X → Y is closed over C ⊂Y if and only if, for eachy∈C and each neighborhoodU off−1(y)inX (which might be empty), there exists a neighborhoodV of y in Y such thatf−1(V)⊂U.
118 2 Fundamental Results on Infinite-Dimensional Manifolds
Proof. To see the “if” part, letAbe a closed set inX. For eachy∈C\f(A), sinceX\Ais a neighborhood of f−1(y), y has a neighborhoodV inY such that f−1(V) ⊂X \A. This implies that V ∩f(A) = ∅, so y ∈ C\clf(A).
Thus, C\f(A) ⊂ C\clf(A), that is, f(A)∩C ⊃ clf(A)∩C. Therefore, f(A)∩C= clf(A)∩C.
To see the “only if” part, let y ∈ C and U be an open neighborhood of f−1(y) in X. Since A=X\U is a closed set inX, we have f(A)∩C = clf(A)∩C. Sincey∈C\f(A)⊂Y\clf(A),V =Y\clf(A) is a neighborhood ofy inY. Then, f−1(V)∩A=∅, that is,f−1(V)⊂X\A=U. ⊓⊔
The following can easily be observed:
Lemma 2.8.10 Let f :X→Y be a map that is closed overC⊂Y. (i) For each closed setAin X,f|A:A→Y is closed overC.
(ii) For eachW ⊂Y,f|f−1(W) :f−1(W)→W is closed over C∩W. ⊓⊔ Proposition 2.8.11 Let f :X →Y be a map of a metrizable space X into an ANRY andAbe a closed set inX such thatf|Ais a closed embedding. For each strong Z-set C inY andU ∈cov(Y),f isU-close to a mapg:X →Y such that f|A=g|A,g(X\A)⊂Y \C, andg is closed overC.
Proof. We can take d∈Metr(X) and ρ∈Metr(Y) such that{Bρ(y,1)|y∈ Y} ≺ U (1.3.18). We inductively construct maps fi :X →Y and closed sets Xiin X, i∈N, so as to satisfy the following:
(1) For each x ∈ X \Xi, there exists a ∈ A such that d(x, a) < 2−i and ρ(fi−1(x), fi−1(a))<2−i,
(2) Xi−1⊂intXi⊂Xi⊂X\A,
(3) ρ(fi, fi−1)<2−i,fi|A∪Xi−1=fi−1|A∪Xi−1and clfi(Xi)∩C=∅, wheref0=f andX0=∅.
Suppose thatfi−1 andXi−1 have been obtained. Let U ={
x∈X d(x, a), ρ(fi−1(x), fi−1(a))<2−i for somea∈A} . Then,U is an open neighborhood ofAin X. Since (X\U)∪Xi−1⊂X\A, we can find a closed setXi inX such that
(X\U)∪Xi−1⊂intXi⊂Xi ⊂X\A.
Then,Xisatisfies conditions (1) and (2). Since clfi−1(Xi−1)∩C=∅, it follows that fi−1(Xi−1)∩N1=∅ for some closed neighborhoodN1of Cin Y. Since C is a strongZ-set inY, we have a 2−i-homotopyh:X×I→Y such that
h0=fi−1, h((Xi−1)×I)∩N1=∅ and clh1(X)∩C=∅.
Then, C has a closed neighborhood N in Y such that h1(X)∩N = ∅ and N ⊂N1, which implies
2.8 Z-Sets and StrongZ-Sets in ANRs 119
A Xi Xi−1
h−1(N) the graph ofα
I
V ×I
Fig. 2.19.A mapα
((Xi−1×I)∪(X× {1}))∩h−1(N) =∅.
Moreover, Xi−1 has an open neighborhood V in X such that (V ×I)∩ h−1(N) = ∅. Take a Urysohn map α : X → I such that α(A∪Xi−1) = 0 and α(Xi\V) = 1. Then, (x, α(x))̸∈h−1(N) for everyx∈Xi. We define a map fi : X →Y by fi(x) = h(x, α(x)). Then, fi satisfies condition (3). — Fig. 2.19.
Now, we haveX\A=∪
i∈NXi by (1) and (2). By virtue of (3), we can defineg:X→Y byg|A=f|Aandg|Xi=fi|Xifor eachi∈N. Since (fi)i∈N
is uniformly Cauchy by (3),g is continuous as the uniform limit of (fi)i∈N. It also follows that ρ(g, fi)<2−i. In particular,ρ(g, f) =ρ(g, f0)<1, hence g isU-close tof. Moreover,
g(X\A) =g( ∪
i∈NXi) =∪
i∈N
fi(Xi)⊂Y \C.
To see thatg is closed overC, assume, on the contrary, that there exists z ∈C with an open neighborhood V of g−1(z) inX such thatg−1(W)̸⊂V for any neighborhoodW ofzinY (Lemma 2.8.9). Then, we have a sequence (xn)n∈NinX\V such that (g(xn))n∈Nconverges toz. Due to (3), for eachi∈ N, there is a neighborhoodWiofzin Y that is disjoint fromfi(Xi) =g(Xi).
This implies that (xn)n∈N has a subsequence (xni)i∈N such that xni ̸∈ Xi
for each i ∈ N. By (1), we can take ai ∈ A, i ∈N, so thatd(xni, ai)<2−i and ρ(fi−1(xni), fi−1(ai)) < 2−i. Since ρ(g, fi−1) < 2−i+1, it follows that ρ(g(xni), g(ai)) < 2−i+3, which means that (g(ai))∈N converges to z. Since g|A = f|A is a closed embedding, it follows that (ai)∈N converges to some a ∈ A. Recall that d(xni, ai) < 2−i, hence (xni)i∈N also converges to some a ∈ A, which means thata ∈ X\V and z = g(a). However, g−1(z) ⊂ V, which is a contradiction. Consequently,g is closed overC. ⊓⊔
In Theorem 2.8.6, it was shown that a closed setAin an ANRXis aZ-set if and only ifX\A is homotopy dense inX. For strong Z-sets, we have the following:
120 2 Fundamental Results on Infinite-Dimensional Manifolds
Proposition 2.8.12 A closed set A in an ANR X is a strong Z-set if and only if there is a homotopyh:X×I→X such that
h0= id, h(X×(0,1])⊂X\A andhis closed over A.
Proof. IfAis a strongZ-set, then we can obtain the desired homotopy hby applying Proposition 2.8.11 to prX:X×I→X andX× {0} ⊂X×I. Thus, we have the “only if” part.
To see the “if” part, suppose the homotopyh, as defined above, is given.
For eachU ∈cov(X), we apply Lemma 2.5.2 to obtain a mapα:X →(0,1]
so that {
{x} ×[0, α(x)]x∈X}
≺h−1(U).
We can define a map f : X → X by f(x) = h(x, α(x)). Then, f is U-close to id. Sincehis closed overA, it follows from Lemma 2.8.10(i) thatf is also closed overA, hence we have A∩clf(X) =A∩f(X) =∅. ⊓⊔
Now, we can show the following:
Proposition 2.8.13 LetAbe a strongZ-set in an ANRX andU be an open set in X. Then, A∩U is a strongZ-set inU.
Proof. By Proposition 2.8.12, we have a homotopyh:X×I→X such that h0 = id, h(X ×(0,1]) ⊂ X \A, and h is closed over A. Then, note that h−1(U) is an open set inX×Iwithh−1(U)∩(X× {0}) =U× {0}. Applying Lemma 2.5.2, we can obtain a map α : X → I such that α(U) ⊂ (0,1]
and {x} ×[0, α(x)] ⊂ h−1(U) for each x ∈ U. We can define a homotopy g:U×I→Ubyg(x, t) =h(x, tα(x)). Then,g0= id andg(U×(0,1])⊂U\A.
It follows from Lemma 2.8.10 thatgis closed overA∩U. By Proposition 2.8.12, A∩U is a strongZ-set inU. ⊓⊔
Remark 2.10 In the above, given U ∈ cov(U), we have a map ˜α: X → I such that ˜α−1(0) =X\U, ˜α|U 6α, and{{x} ×[0,α(x)]˜ |x∈U} ≺ U. We define ˜g:X×I→X by ˜g(x, t) =h(x, t˜α(x)). Then, ˜gis aU-homotopy such that ˜g0= id, ˜g(U×(0,1])⊂U\A, and ˜gt|X\U = id for everyt∈I. Moreover,
˜
g|U×Iis closed overA∩U, hence cl ˜g1(U)∩A∩U = ˜g1(U)∩A∩U =∅.17 To prove the following proposition, we can apply Michael’s Theorem 1.3.16 on local properties.
Proposition 2.8.14 A closed set Ain an ANR X is a strongZ-set inX if each a ∈ A has an open neighborhood U in X such that A∩U is a strong Z-set inU.
Proof. To apply Michael’s Theorem 1.3.16, it is enough to prove the following:
17We can also apply Proposition 2.1.16 to obtain such a homotopy.