Homology, Homotopy and Applications, vol.5(1), 2003, pp.213–218

P-EMBEDDINGS, AR AND ANR SPACES

L. STRAMACCIA

(communicated by Walter Tholen)

Abstract Generalizations of Borsuk’s characterization theorems for AR and ANR spaces (for metrizable spaces) are provided. Moreover, a simple proof of Morita’s theorem is given, in generalization of Borsuk’s homotopy extension theorem.

Introduction

The classes of absolute retracts 𝒜ℛ and of absolute neighborhood retracts 𝒜𝒩ℛ (for metrizable spaces) where introduced by K. Borsuk. They play an important role in Homotopy Theory and in Algebraic Topology essentially because, in the category ℳ of metrizable spaces, 𝒜ℛ is the class of injective objects with respect to closed embeddings, while ANR-spaces are homotopically equivalent to CW-spaces and to polyhedra. Borsuk’s theorems characterizing 𝒜ℛ and 𝒜𝒩ℛ ([3], Thm. 4.2, p. 87) are concerned with closed pairs of metrizable spaces. We generalizes them to pairs (X, B), where X is an arbitrary topological space and B is a P-embedded ([2]) zero-set of X. Borsuk’s homotopy extension theorem ([3], Thm. 8.1, p. 94) asserts that every closed pair (X, B) of metrizable spaces has the homotopy extension property with respect to the class 𝒜𝒩ℛ. The latest and most remarkable generalization of this theorem is due to Morita ([11]). He showed that the same result holds for every pair (X, B), where B is a P-embedded, zero-set of X. The definition of P-embedded subset used there is different from the previous one but the two are equivalent as shown in [8]. We were led to rediscover such a result, for which we present a new and simple proof. ℳ is a proreflective subcategory of the category 𝒯 of all topological spaces, hence every space can be approximated by an inverse system of metrizable spaces (see Section 1). This fact allows one to extend Borsuk’s original proofs from closed pairs in ℳ to general pairs (X, B) in 𝒯, where B is a P-embedded zero-set of X.

1. Preliminaries

Let 𝒞 be a fixed category. In the following we shall work within the category Pro𝒞 of inverse systems in 𝒞. We refer to [10] for all details of its definition. We only

Received October 24, 2002, revised June 17, 2003; published on June 30, 2003. 2000 Mathematics Subject Classification: 55P05, 54E35, 54C20, 54C56, 18A30. Key words and phrases: continuous pseudometric, metric expansion, proreflection, P-embedding, absolute (neighborhood) retract, homotopy extension property. \begin{matrix}c \\ ◯ \end{matrix} 2003, L. Stramaccia. Permission to copy for private use granted.

recall the following basic facts that will be explicitly used in the sequel :

- an inverse system in 𝒞 is a contravariant functor X : Λ → 𝒞, where Λ = (Λ, ⩽) is a directed set, and will be denoted as X = (X_{λ}, x_{λλ′}, Λ), where X_{λ} = X(λ) , for every λ ∈ Λ, and x_{λλ′} : X_{λ′} → X_{λ} , where x_{λλ′} = X(λ ⩽ λ′).

- a morphism p : X → X, where X ∈ 𝒞, is simply a cone, i.e. a family p = {p_{λ} : X → X_{λ} | λ ∈ Λ} of morphisms of 𝒞 having the property that x_{λλ′} · p_{λ′} = p_{λ} , for all λ ⩽ λ′ .

- if 𝒦 is a full subcategory of 𝒞 then Pro𝒦 is a full subcategory of Pro𝒞. 𝒦 is said to be proreflective in 𝒞 whenever Pro𝒦 is reflective in Pro𝒞. This amounts to saying ([4], [12]) that, for every X ∈ 𝒞, there exists an inverse system X ∈ Pro𝒦 and a morphism p : X → X in Pro𝒞 having the following properties :

(1) for every f : X → K, K ∈ 𝒦, there exists a λ ∈ Λ and a morphism f_{λ} : X_{λ} → K, such that f_{λ} · p_{λ} = f.

(2) if f_{λ′} : X_{λ′} → K has also the property that f_{λ′} · p_{λ′} = f, then there exists a λ” ∈ Λ, λ, λ′ ⩽ λ”, such that f_{λ} · x_{λλ”} = f_{λ′} · x_{λ′λ”}. If this is the case, then p : X → X is said to be a 𝒦-expansion for X. It is clear that a 𝒦-expansion for X is uniquely determined up to isomorphisms in Pro𝒦.

A subcategory 𝒦 of the category 𝒯 of topological spaces (resp. 𝒯_{2} of Hausdorff spaces) is proreflective if and only if [6] it is closed under finite products and subspaces (resp. closed subspaces). With this result at hand, one can produce a great number of proreflective subcategories of topological spaces. In particular, the categories 𝒫ℳ and ℳ of pseudometrizable and metrizable spaces, respectively, are both proreflective in 𝒯. In what follows we discuss these facts.

Let X = (X, τ) be a topological space. Recall that a pseudometric on X is a map λ : X × X → ℝ^{+} having the following properties:

(i) λ(x, x) ⩾ 0,

(ii) λ(x, y) = λ(y, x),

(iii) λ(x, y) + λ(y, z) ⩽ λ(x, z),

for all x, y, z ∈ X. λ is said to be continuous [2] if it is continuous with respect to the product topology on X × X. If τ_{λ} denotes the topology induced on X by the pseudometric λ, then λ will be continuous if and only if τ_{λ} ⊂ τ holds.

Let Λ be the set of all continuous pseudometrics on X, modulo the relation that identifies two pseudometrics whenever they induce the same topology on X. Λ is directed by the relation λ ⩽ λ′ ⟺ τ_{λ} ⊂ τ_{λ′}.

For every λ ∈ Λ, let X_{λ} denote the metric identification of the pseudometric space (X, τ_{λ}) while, for λ ⩽ λ′, x_{λλ′} : X_{λ′} → X_{λ} is the map induced on the quotients by the the identity (X, τ_{λ′}) → (X, τ_{λ}) , which is continuous. Then X = (X_{λ}, x_{λλ′}, Λ) is an inverse system in the category ℳ and the morphism x : X → X where, for λ ∈ Λ, x_{λ} : X → X_{λ} is the identity (X, τ) → (X, τ_{λ}) followed by the identification map, turns out to be a ℳ-expansion for X [6], [7]. We call it the metrizable expansion of X.

Recall from [2] that a subset B of a topological space X is said to be P -embedded in X if every continuous pseudometric on B can be continuously extended over X. An equivalent formulation for this definition is given in [8], Thm. 2.10.

Note that, whenever B is P-embedded in X, then the metrizable expansion b : B → B of B is obtained by restriction from that of X. In fact, in such a case every continuous pseudometric on B can be thought of as the restriction of any of its extensions to X. The embedding i : B → X induces a level morphism, actually a natural transformation of functors, i : B → X, i = {i_{λ} : B_{λ} → X_{λ} | λ ∈ Λ}. That is, for every λ ∈ Λ, there is a commutative diagram

[figure: B; B_{λ}; X; X_{λ}; b_{λ}; x_{λ}; i; i_{λ}]

with i_{λ} the corresponding inclusion.

Proposition 1.1. Let X be a topological space and let x : X → X be its metrizable expansion. For a P-embedded subset B of X the following properties are equivalent:

(1) B is a zero-set of X.

(2) For every μ ∈ Λ, there exists λ ∈ Λ, λ ⩾ μ, such that B_{λ} is closed in X_{λ}.

Proof. Let B be a P-embedded, zero-set of X. There exists a continuous map f : X → ℝ such that B = f^{−1}(0). From the universal property of the metrizable expansion of X one obtains, for some η ∈ Λ, a map f_{η} : X_{η} → ℝ such that f = f_{η}·x_{η}. It follows B = f^{−1}(0) = x_{η}^{−1}f_{η}^{−1}(0), hence B_{η} = x_{η}(B) = f_{η}^{−1}(0). For any μ ∈ Λ, let λ ⩾ η, μ, then B_{λ} = f_{λ}^{−1}(0), for f_{λ} = f_{η} · x_{ηλ}. Let now B be a P-embedded subset of X for which (2) holds and let λ ∈ Λ be any continuous pseudometric on X such that B_{λ} is closed in X_{λ}. Here B_{λ} denotes the pseudometrizable subspace of X_{λ} defined by B. Then B_{λ} is a zero-set of X_{λ}, hence there exists a continuous map f_{λ} : X_{λ} → ℝ with B_{λ} = f_{λ}^{−1}(0). It follows that B = (f_{λ} · x_{λ})^{−1}(0).

In the following we shall say that a subset B of a space X is P_{0}-embedded if it is P-embedded and satisfies the equivalent conditions of the above proposition.

Examples 1.2.

1. Every closed subset B of a metrizable space X is P_{0}-embedded. This follows from Arens’ theorem ([2], 15.4), which states that every closed subset of a metrizable space is P-embedded.

2. The P_{0}-embedded subsets of a collectionwise normal space are exactly its zero-sets ([2], 15.7).

3. Every closed, P-embedded, G_{δ}-subset of a normal space is P_{0}-embedded.

4. Every closed, P-embedded subset of a perfectly normal T_{1}-space is P_{0}-embedded.

5. Every C-embedded zero-set of a normal, countably paracompact space is P_{0}- embedded.

2. Results

The machinery we have given in the previous section allows us to obtain the following generalization of Borsuk’s characterization theorems for AR and ANR spaces.

Theorem 2.1. Let Y be a metrizable space. Then

(1) Y ∈ 𝒜ℛ if and only if, for every pair (X, B) of topological spaces, where B is P_{0}-embedded in X, every map f : B → Y has a continuous extension f̃: X → Y.

(2) Y ∈ 𝒜𝒩ℛ if and only if, for every pair (X, B) of topological spaces, where B is P_{0}-embedded in X, every map f : B → Y has a continuous extension f̃: U → Y, to some neighborhood U of B in X.

Proof. (1) Let Y ∈ 𝒜ℛ and let f : B → Y be any map. Since Y is metrizable, by the universal property of the metrizable expansion b : B → B, one can find λ ∈ Λ and a map g_{λ} : B_{λ} → Y such that g_{λ} · b_{λ} = f. Since B is P_{0}-embedded in X, then one can assume that B_{λ} is closed in the metrizable space X_{λ}. Applying now the first theorem of Borsuk to the pair (X_{λ}, B_{λ}), there exists a map h_{λ} : X_{λ} → Y which extends g_{λ}, that is h_{λ} · i_{λ} = g_{λ}. Finally, f = g_{λ} · b_{λ} = h_{λ} · i_{λ} · b_{λ} = h_{λ} · x_{λ} · i, so that f̃ = h_{λ} · x_{λ} is an extension of f. Conversely, let every map f : B → Y have an extension f̃ : X → Y, whenever B is P_{0}-embedded in X. Since this holds in particular for every closed pair (X, B) of metrizable spaces, from Borsuk’s theorem it follows that P has to be an AR.

(2) The proof of the second part is analogous to that of the first one, unless to take an extension h_{λ} : U_{λ} → Y of g_{λ}, for U_{λ} a neighborhood of B_{λ} in X_{λ}, whose existence is guaranteed by the second Borsuk theorem. Then consider U = x_{λ}^{−1}(U_{λ}) to obtain the extension f̃ = h_{λ} · x_{λ} : U → Y of f.

Recall that a continuous map i : B → X is said to have the homotopy extension property (HEP) with respect to a class 𝒦 of topological spaces, whenever the diagram

[figure: B; B × I; X; X × I; e_{0}^{B}; e_{0}^{X}; i; i × 1]

is a weak pushout with respect to 𝒦. This means that, for every map g : X → Y, Y ∈ 𝒦, and homotopy F : B × I → Y such that F · e_{0}^{B} = g · i, there exists a homotopy G : X × I → Y with G · e_{0}^{X} = g and G · (i × 1) = F. Here e_{0}^{X} : X → X × I is the map defined by e_{0}^{X}(x) = (x, 0), for all x ∈ X. In the sequel we will consider the HEP for a pair (X, B) of topological spaces, where B is a subset of X and i : B → X is the embedding map.

Theorem 2.2. (The Homotopy Extension Theorem) A pair of topological spaces (X, B) has the homotopy extension property with respect to 𝒜𝒩ℛ whenever B is P_{0}-embedded in X.

Proof. Given a commutative diagram

[figure: B; B × I; X; Y; e_{0}^{B}; G_{0}; i; F]

where Y ∈ 𝒜𝒩ℛ, we have to construct an extension G : X × I → Y of F, such that G · e_{0}^{X} = G_{0}. Note that the metrizable expansion of X × I is given by x × 1 : X × I → X × I = (X_{λ}×I, x_{λλ′} ×1, Λ), and the related metrizable expansion of B×I by b×1 : B×I → B × I = (B_{λ} × I, b_{λλ′} × 1, Λ). By the proreflection property, there exists an index λ ∈ Λ and a homotopy F^{λ} : B_{λ} × I → Y, such that F^{λ} · (b_{λ} × 1) = F. On the other hand, there exists a λ′ ∈ Λ and a map G_{0}^{λ′} : X_{λ} → Y, such that G_{0}^{λ′} · x_{λ′} = G_{0}. Since the equality G_{0}^{λ′} · i_{λ′} · b_{λ′} = F_{λ} · e_{0}^{B_{λ}} · b_{λ} = G_{0} · i holds, then there exists a λ^{∗} ⩾ λ, λ′ such that F_{λ} · e_{0}^{B_{λ}} · b_{λλ^{∗}} = G_{0}^{λ′} · i_{λ′} · b_{λ′λ^{∗}}. Since B is P_{0}-embedded in X we can assume that, for such a λ^{∗}, the corresponding embedding i_{λ^{∗}} is closed. Let us define G_{0}^{λ^{∗}} = G_{0}^{λ′} · x_{λ′λ^{∗}} and F^{λ^{∗}} = F^{λ} · (b_{λλ^{∗}} × 1). Then the following diagram is commutative

[figure: B_{λ^{∗}}; B_{λ^{∗}} × I; X_{λ^{∗}}; Y; ,; e_{0}^{B_{λ}^{∗}}; G_{0}^{λ^{∗}}; i_{λ^{∗}}; F^{λ^{∗}}]

in fact, F^{λ^{∗}} · e_{0}^{B_{λ^{∗}}} = F^{λ} · e_{0}^{B_{λ}} · b_{λλ^{∗}} = G_{0}^{λ′} · i_{λ′} · b_{λ′λ^{∗}} = G_{0}^{λ′} · x_{λ′λ^{∗}} · i_{λ^{∗}} = G_{0}^{λ^{∗}} · i_{λ^{∗}}. If we apply now Borsuk’s homotopy extension theorem to the pair (X_{λ^{∗}}, B_{λ^{∗}}), we get the extension G^{λ^{∗}} : X_{λ^{∗}} × I → Y of F^{λ^{∗}}. Finally G = G^{λ^{∗}} · (x_{λ^{∗}} × 1) turns out to be the extension of F we were searching for.

References

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This article may be accessed via WWW at http://www.rmi.acnet.ge/hha/ or by anonymous ftp at

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L. Stramaccia stra@dipmat.unipg.it

Dipartimento di Matematica e Informatica Universitá di Perugia via Vanvitelli, I-06123 Perugia