Theorem 1.7.5 A countable-dimensional metrizable spaceX with dimX =
∞is weakly infinite-dimensional. Equivalently, any strongly infinite-dimensional metrizable space is not countable-dimensional. ⊓⊔
The following will be used in Chapters 4 and 5:
[5.8.1]
Theorem 1.7.6 (Embedding Theorem) Every separable metrizable space with dim 6 n can be embedded in I2n+1, hence it can be embedded in the Euclidean space R2n+1.
1.8 ANRs
A subsetAof a spaceX is called aretractofX if there is a mapr:X →A such thatr|A= id, which is called aretraction. As is easily observed, every retract of a spaceX isclosedinX. Aneighborhood retractofX is aclosed set inX that is a retract of some neighborhood in X. Ametrizable spaceX is called an absolute neighborhood retract (ANR) (resp. an absolute retract (AR)) ifX is a neighborhood retract (or a retract) of an arbitrary metrizable space containing X as a closed subspace. A space Y is called an absolute neighborhood extensor for metrizable spaces(ANE) if each map f : A → Y from any closed set A in an arbitrary metrizable space X extends over some neighborhoodU ofAinX. Whenf can always be extended over X (i.e., U = X in the above), we call Y an absolute extensor for metrizable spaces(AE). As is easily observed, everymetrizableANE (resp.
a metrizable AE) is an ANR (resp. an AR). Moreover, every neighborhood retract of an ANE is an ANE and every retract of an AE is an AE.
The following theorems are foundations of ANR theory.
[6.1.1]
Theorem 1.8.1 (Dugundji Extension Theorem) LetEbe a locally con-vex topological linear space and X be a metrizable space with a closed set A ⊂X. Then, each map f :A → E can be extended to a map f˜: X →E such that the image f˜(X) is contained in the convex hull ⟨f(A)⟩ of f(A), hence every convex set is an AE.
Remark 1.4 In the above, whenE= (E,∥ · ∥) is a normed linear space and f is bounded, the extension ˜f can be taken so as to be bounded and have the same sup-norm asf, (i.e., ∥f˜∥=∥f∥). Cf. Remark 2 in [GAGT, Chap.6].
[6.2.1]
Theorem 1.8.2 (Arens–Eells Embedding Theorem) Every (complete) metric space X can be isometrically embedded in a (complete) normed linear spaceE with densE=ℵ0densX as a linearly independent closed set.
By combining these two theorems, we can obtain the following:
• ametrizablespaceX is an ANR (resp. an AR) if and only ifX is an ANE (resp. an AE).
36 1 Preliminaries and Background Results Moreover, the following holds:
[6.2.9] Proposition 1.8.3 A metrizable space is an AR if and only if it is a con-tractible ANR.
The following is a very useful procedure to extend homeomorphisms:
[6.2.2] Theorem 1.8.4 (Klee’s Trick) LetE andF be metrizable topological lin-ear spaces which are AEs and let AandB be homeomorphic closed sets inE andF, respectively. Then, each homeomorphismf :A× {0} → {0} ×B can be extended to a homeomorphismf˜:E×F→E×F.
The Klee’s trick can be applied to prove the following Hausdorff’s Metric Extension Theorem:
[6.2.3] Theorem 1.8.5 (Hausdorff’s Metric Extension) LetAbe a closed set in a (completely) metrizable spaceX. Every admissible (complete) metric on A extends to an admissible (complete) metric onX.
[6.2.4] Theorem 1.8.6 Every completely metrizable space can be embedded in a Hilbert space with the same density as a closed set. And every metrizable space can be embedded in a pre-Hilbert space (that is, a linear subspace of a Hilbert space) with the same density as a closed set.
Proposition 1.8.7 (Basic Properties of ANEs) [6.1.9]
(1) An arbitrary product of AEs is an AE and a finite product of ANEs is an ANE.
(2) A retract of an AE is an AE and a neighborhood retract of an ANE is an ANE.
(3) Any open set in an ANE is also an ANE.
(4) (Hanner Theorem) A paracompact space is an ANE if it is locally an ANE, that is, each point has an ANE neighborhood.
(5) Let X = X1∪X2, where X1 and X2 are closed in X. If X1, X2, and X1∩X2 are ANEs (AEs), then so is X. If X and X1∩X2 are ANEs (AEs), then so areX1 andX2.
Proposition 1.8.8 (Basic Properties of ANRs) [6.2.10]
(1) An countable product of ARs is an AR and a finite product of ANRs is an ANR.
(2) A retract of an AR is an AR and a neighborhood retract of an ANR is an ANR.
(3) Any open set in an ANR is also an ANR.
(4) (Hanner Theorem) A paracompact space is an ANR if it is locally an ANR, that is, each point has an ANR neighborhood.
1.8 ANRs 37 (5) Let X = X1∪X2, where X1 and X2 are closed in X. If X1, X2, and X1∩X2 are ANRs (ARs), then so is X. If X and X1∩X2 are ANRs (ARs), then so areX1 andX2.
[6.1.4]
[6.2.6]
Theorem 1.8.9 For any simplicial complexK, the polyhedron|K|is an ANE and|K|m is an ANR.
[6.4.1]
Theorem 1.8.10 (Homotopy Extension Theorem) Let Y be an ANE, U be an open cover of Y, andh:A×I→Y be aU-homotopy of a closed set Ain a metrizable space X. Ifh0is extended to a mapf :X →Y, then hcan be extended to aU-homotopyh˜:X×I→Y with ˜h0=f.
LetV be an open refinement of an open cover U of a spaceX. We callV an h-refinement(resp.~~~-refinement) ofU if any twoV-close mapsf, g : Y →X defined on an arbitrary spaceY areU-homotopic (resp.U-homotopic rel.{y∈Y |f(y) =g(y)}), where we write
V ≺h U or U ≻
h V (
resp.V ≺
~ U or U ≻
~ V) .
[6.3.5]
Theorem 1.8.11 Every open cover of an ANR has an~-refinement (hence it has anh-refinement).
[6.5.2]
Theorem 1.8.12 (Kruse–Liebnitz) LetX be metrizable andAbe a strong neighborhood deformation retract of X. IfA andX\Aare ANRs, then so is X. ⊓⊔
For a mapf :X→Y, we define themapping cylinderM(f) as the space Y∪(X×(0,1]) (the disjoint union) with the topology generated by open sets inX×(0,1] and setsV∪f−1(V)×(0, ε), whereV is open inY and 0< ε <1.
The adjunction spaceMf =Y ∪fprX|X×{0}X×Iis also called themapping cylinder. If X andY are metrizable, thenM(f) is also metrizable, but Mf
is not in general. Then, we callM(f) themetrizable mapping cylinderto distinguish it from Mf. It should be noted that if X and Y are completely metrizable, then so isM(f). For a perfect mapf :X→Y between metrizable spaces, Mf is metrizable andMf ≈M(f). — For these facts, refer to Sect.
6.5 (pp.362–3) in [GAGT].
As an easy application of the Kruse–Liebnitz Theorem 1.8.12 above, we have the following:
[6.5.4]
Corollary 1.8.13 For any map f : X → Y between ANRs, the mapping cylinderM(f)is an ANR. ⊓⊔
The (metrizable) cone C(X) (resp. the (metrizable) open cone Co(X)) over a (metrizable) spaceX is defined as the space{0} ∪X×(0,1]
(resp. Co(X) = {0} ∪X×(0,1)) with the topology generated by open sets in X ×(0,1] (resp. X ×(0,1)) and sets {0} ∪X ×(0, ε), 0 < ε < 1. In
38 1 Preliminaries and Background Results
other words, the cone C(X) is the mapping cylinderM(c0) of the constant map c0 : X → {0}. The quotient space (X ×I)/(X × {0}) is also called the cone over X, which is another mapping cylinder Mc0 of the constant map c0 : X → {0}. If X is metrizable, then C(X) is also metrizable but (X×I)/(X× {0}) is not in general. Moreover, ifX is completely metrizable, then so areC(X) andCo(X). WhenXis compact, (X×I)/(X×{0})≈C(X).
The following is the special case of Corollary 1.8.13 above:
[6.5.7] Corollary 1.8.14 The cone C(X) over any ANR X is an AR, hence so is the open coneCo(X). ⊓⊔
The following is called Hanner’s characterization of ANRs:
[6.6.2] Theorem 1.8.15 (Hanner) For a metric space X = (X, d), the following are equivalent:
(a) X is an ANR;
(b) For each open coverU of X, there is a simplicial complex K such thatX isU-homotopy dominated by|K|;
(c) For anyε >0,X isε-homotopy dominated by an ANE.
Remark 1.5 In condition (b) above, we can require K to be locally finite-dimensional and cardK(0) 6w(X). — Refer Remark 9 for Theorem 6.6.2 in [GAGT].
[6.6.3] Theorem 1.8.16 Let f :X →Y be a map from a paracompact space X to an ANRY andU be an open cover of Y. Then, each open coverV of X has an open refinement W with a map ψ :|N(W)| → Y such that ψφ≃U f for any canonical mapφ:X→ |N(W)|. ⊓⊔
Remark 1.6 In the above theorem, we can take a locally finite σ-discrete open refinementWofVsuch that the nerveN(W) is locally finite-dimensional.
WhenX is separable, we can take a star-finite countable open refinementWof Vsuch that the nerveN(W) is locally finite. — Refer Remark 10 for Corollary 6.6.3 in [GAGT].
A subsetAof a spaceX said to behomotopy denseinX if there exists a homotopyh:X×I→X such thath0= id andh(X×(0,1])⊂A.
[6.6.7] Theorem 1.8.17 Let X be a metrizable space and A be a homotopy dense subset ofX. Then,X is an ANR if and only ifA is an ANR. ⊓⊔
[6.6.8] Proposition 1.8.18 Let x0 ∈A⊂X. IfA is homotopy dense in X and X is contractible, then the following setANf is homotopy dense in XN:
ANf ={
x∈ANx(n) =x0 except for finitely manyn∈N}
.
1.8 ANRs 39 A map f :X →Y is called a weak homotopy equivalence provided that, for each n∈ Nand each mapα :Sn−1 →X, iff α extends to a map β : Bn →Y, then αextends to a map ˜α:Bn →X such thatfα˜ ≃β in Y rel. Sn−1. A mapf :X →Y is a weak homotopy equivalence if and only if f induces the bijection between the path-components and the isomorphisms
between the homotopy groups (cf. Theorem 4.14.12 in [GAGT]). [4.14.12]
[6.6.6]
Theorem 1.8.19 LetX andY be ANRs. Then, every weak homotopy equiv-alencef :X→Y is a homotopy equivalence. ⊓⊔
[6.6.4]
Theorem 1.8.20 Every ANR X has the homotopy type of a locally finite-dimensional simplicial complexKwithcardK(0)6w(X). In particular, every separable ANR has the homotopy type of a countable locally finite simplicial complex. ⊓⊔
LetX = (X, dX) andY = (Y, dY) be metric spaces andAbe a closed set inX. A mapf :X →Y is said to beuniformly continuousatAif, for each ε >0, there is someδ >0 such that for eacha∈A andx∈X,dX(a, x)< δ impliesdY(f(a), f(x))< ε. We call Aa uniform retract ofX if there is a retractionr:X →Athat is uniformly continuous atA. A subsetU ⊂X such that dist(A, X\U)>0 is called auniform neighborhoodofAinX. When Ahas a uniform neighborhood U inX andA is a uniform retract ofU,Ais called auniform neighborhood retractofX. As is easily observed, every compact (neighborhood) retract ofX is a uniform (neighborhood) retract.
A metric spaceX is called a uniform AR (resp.uniform ANR) if X is a uniform retract (resp. a uniform neighborhood retract) of an arbitrary metric space containingX as a closed set.
[6.8.5]
Theorem 1.8.21 Every convex set in a locally convex metric linear space is a uniform AE, hence a uniform AR. ⊓⊔
[6.8.8]
Proposition 1.8.22 A uniform ANR X is homotopy dense in every metric spaceZ that containsX isometrically as a dense subset.
[6.8.10]
Theorem 1.8.23 A metric spaceX is a uniform ANR (resp. a uniform AR) if and only if the metric completionXe ofXis a uniform ANR (resp. a uniform AR) and X is homotopy dense inXe. ⊓⊔
[6.8.2]
Proposition 1.8.24 Let X be a subset of a metric spaceM = (M, d). If X is an ANR, then X is homotopy dense in some Gδ-set Y in M, hence Y is also an ANR.
[6.8.11]
Theorem 1.8.25 For any admissible metric d on an AR (resp. ANR) X, X has an admissible metric ρ >d such that (X, ρ) is a uniform AR (resp.
uniform ANR). Ifdis bounded, then so isρ.
40 1 Preliminaries and Background Results