Georgian Mathematical Journal Volume 10 (2003), Number 2, 201–207

ON ARU-RESOLUTIONS OF UNIFORM SPACES

V. BALADZE

Abstract. In this paper theorems which give conditions for a uniform space to have an ARU-resolution are proved. In particular, a finitistic uniform space admits an ARU-resolution if and only if it has trivial uniform shape or it is an absolute uniform shape retract.

2000 Mathematics Subject Classification: 54C56, 54E15.

Key words and phrases: Uniform shape, uniform resolution, absolute uniform shape retract.

1. Introduction

The definition of resolution of topological spaces was introduced by S. Mardešic [8] and it was used to define and study the shapes of topological spaces [10]. More recently, J. Segal, S. Spiez and B. Günter [14] defined a uniform version of resolution and proved that every finitistic uniform space has ANRU-resolution. They and T. Miyata [11] showed its usefulness in uniform shape theory.

The natural question is to determine which uniform spaces admit ARU-reso-lutions, i.e. uniform resolutions p : X → X = (X_{λ}, p_{λλ′}, Λ), where each X_{λ} is an ARU-space. In [5] A. Koyama, S. Mardešic and T. Watanabe proved that a topological space X has an AR-resolution if and only if X has trivial shape.

In this paper we prove that for every finitistic uniform space X the following assertions are equivalent:

(i) X admits an ARU-resolution.

(ii) X has trivial uniform shape.

(iii) X is an absolute uniform shape retract.

Without specific references we use results from [4], [10], [11] and [14]. A uniform space X is said to be an A(N)RU-space if whenever X is embedded in a uniform space Y, then there is a uniform retraction of Y (some uniform neighborhood of X in Y) onto X, equivalently, if whenever A is a uniform subspace of Y, then every uniform map f : A → X extends over Y (some uniform neighborhood of A in Y). We say that two uniform maps f, g : X → Y are semi-uniformly homotopic maps and write f ≃_{u} g if there exists a uniform map H : X ∗ I → Y such that H_{0} = f and H_{1} = g. Here X ∗ I is the semi-uniform product of X and I = [0, 1] (see [4, p. 44]).

A uniform space X is said to be semi-uniform contractible [11] if the identity uniform map 1_{X} : X → X and some constant uniform map c : X → X are semi-uniformly homotopic maps. It is clear that a uniform space is semi-uniform

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contractible if and only if it has the semi-uniform homotopy type of a uniform space consisting of one point.

Let Unif denote the category of uniform spaces and uniform maps. By A(N)RU we denote the full subcategory of Unif whose objects are A(N)RUspaces. Also, we write H(Unif) and H(A(N)RU) for the semi-uniform homotopy category of Unif and A(N)RU, respectively.

There have been uniform shape theories defined by many mathematicians ([1], [2], [3], [12], [13]), based on the uniform homotopy [4]. Recently, T. Miyata [11] introduced a new uniform shape theory by the inverse system approach, using the semi-uniform homotopy which is weaker than the uniform homotopy. Here we consider T. Miyata’s uniform shape theory which is convenient for our purpose. The uniform shape category uSh is the abstract shape category Sh_{(T ,P)} [10], where T = H(Unif) and P = H(ANRU). The uniform shape morphism F : X → Y of category uSh is given by a triple (p, q, f), where p : X → X = (X_{λ}, p_{λλ′}, Λ) ∈ pro − H(ANRU) and q : Y → Y = (Y_{μ}, q_{μμ′}, M) ∈ pro − H(ANRU) are H(ANRU)-expansions of X and Y, respectively, and f : X → Y is a morphism of category pro−H(ANRU). Two uniform spaces X and Y are said to have the same uniform shape denoted by ush(X) = ush(Y), if X and Y are isomorphic objects of category uSh. We say that a uniform space X has trivial uniform shape and write ush(X) = 0 if X has the uniform shape of one point {∗}.

2. Aru-Resolution

A uniform resolution p = (p_{λ}) : X → X = (X_{λ}, p_{λλ′}, Λ) of a uniform space X consists of an inverse system X in Unif and of a morphism p in pro − Unif, which has the property that for every ANRU-space P and every uniform covering V of P the following two conditions are satisfied:

(UR1) For every uniform map f : X → P there exist λ ∈ Λ and a uniform map h : X_{λ} → P such that the maps h · p_{λ} and f are V-near, i.e. every x ∈ X admits V ∈ V such that h · p_{λ}(x), f(x) ∈ V.

(UR2) There exists a uniform covering V′ of P such that if for λ ∈ Λ and for two uniform maps h, h′ : X_{λ} → P the maps h · p_{λ}, h′ · p_{λ} are V′-near, then there exists λ′ ≥ λ such that the maps h · p_{λλ′}, h′ · p_{λλ′} are V-near.

A uniform resolution is said to be cofinite if its index set Λ is cofinite. If all the terms X_{λ}, λ ∈ Λ, are ANRU(ARU)-spaces, then we speak of an ANRU(ARU)- resolution.

In [14] it is proved that every finitistic uniform space admits an ANRUresolution [14, Theorem 1]. A. Koyama, S. Mardešić and T. Watanabe raised the question of determining which spaces admit an AR-resolution, i.e. a resolution where each term X_{λ} is an absolute retract for metric spaces [5]. In this section we prove

Theorem 1. Let X be a uniform space. Then the following assertions hold:

(i) If X admits an ARU-resolution, then X has trivial uniform shape.

(ii) If X is a finitistic space and has trivial uniform shape, then X admits an ARU-resolution.

The proof of this theorem is based on the following lemmas.

Lemma 2. Let X = (X_{λ}, p_{λλ′}, Λ) and Y = (Y_{μ}, q_{μμ′}, M) be the inverse systems of uniform spaces. If all Y_{μ} are ARU-spaces, then any two morphisms (f_{μ}, φ), (f′_{μ}, φ′) : X → Y of category inv − Unif are equivalent in the sense of morphisms [10].

Proof. For each index μ ∈ M there exists an index λ ∈ Λ such that λ ≥ φ(μ), φ′(μ). Let Z = X_{λ} × {0} ∪ X_{λ} × {1}. Let h : Z → Y_{μ} be a map defined by the formulas

h_{|X_{λ}×{0}} = f_{μ} · p_{φ(μ)λ}, h_{|X_{λ}×{1}} = f′_{μ} · p_{φ′(μ)λ}.

It is clear that h is a uniform map. This map has a uniform extension h′ : X_{λ} ∗ I → Y_{μ}. Hence, f_{μ} · p_{φ(μ)λ} and f′_{μ} · p_{φ′(μ)λ} are semi-uniformly homotopic maps. Consequently, (f_{λ}, φ) ∼ (f′_{λ}, φ′). □

Lemma 3. Let f : X → Y be a uniform map from a uniform space X into an ANRU-space Y. If the map f is semi-uniformly homotopic to a constant map, then there exist an ARU-space Z and a uniform map r : Z → Y such that X ⊆ Z and r_{|X} = f.

Proof. Consider X as a uniform subspace of an ARU-space Z. Since f is semiuniform homotopic to a constant map c : X → Y and since c has a uniform extension on Z, the semi-uniform homotopy extension theorem (see [Theorem 1.6][11]) proves that f : X → Y too extends to a uniform map r : Z → Y. □

Lemma 4. If a semi-uniform contractible space Y is an ANRU-space, then Y is an ARU-space.

Proof. By the assumption of the lemma, there exist a point y_{0} ∈ Y and a semiuniform homotopy H : Y ∗ I → Y such that H(y, 0) = y and H(y, 1) = y_{0} for every y ∈ Y.

Let f : A → Y be a uniform map defined on a uniform subspace A of a uniform space X. Since Y is an ANRU-space, f has a uniform extension f′ : U → Y over some uniform neighborhood U of A in X. There is a uniform covering U of X such that St(A, U) ∩ St(X\U, U) = ∅. By the uniform version of Urysohn’s lemma there exists a uniform map e : X → I such that

e(x) = \begin{cases}0, & x ∈ A, \\ 1, & x ∈ X\St(A, U). \end{cases}

Then we define a map g : X → Y by

g(x) = \begin{cases}H(f′(x), e(x)), & x ∈ U, \\ y_{0}, & x ∈ X\St(A, U). \end{cases}

Let V be an arbitrary uniform covering of X. There exists a uniform covering W of X which is a refinement of the intersection U ∧ V of uniform coverings U and V. Let W ∈ W be an element of W such that W ∩ U ≠ ∅ and W ∩ (X\St(A, U)) ≠ ∅. Now show that W ∩ (U ∩ (X\St(A, U))) ≠ ∅. First assume that W ∩ (X\U) = ∅. Then W ⊂ U and, consequently, W ∩ (U ∩ (X\St(A, U))) = W ∩ (X\St(A, U)) ≠ ∅. Now assume that W ∩ (X\U) ≠ ∅. There exists an element G ∈ U such that W ⊂ G. Note that G ∩ (X\U) ≠ ∅. Hence, G ⊂ St(X\U, U). Since St(A, U) ∩ St(X\U, U) = ∅, we have G ∩ St(A, U) = ∅. Consequently, G ⊂ X\St(A, U). We obtained that G ∩ (X\St(A, U)) = G. Note that W ∩ (X\St(A, U)) = W. This proves that W ∩ (U ∩ (X\St(A, U))) = W ∩ U ≠ ∅.

By Lemma 7 of [6] the map g is a uniform map. For each a ∈ A we have

g(a) = H(f′(a), e(a)) = H(f′(a), 0) = f′(a) = f(a),

i.e. g is an extension of a uniform map f : A → Y. Hence Y ∈ ARU. □

Proof of Theorem 1. We can easily prove the first assertion by following the argument of (i)⇒(ii) in [5]. Here we use our Lemma 2 in the place of [5, Lemma 1]. Indeed, by the assumption (i) X admits an ARU-resolution p = (p_{λ}) : X → X = (X_{λ}, p_{λλ′}, Λ). The space Y = {∗} consisting of one point ∗ also admits an ARU-resolution q = (q_{μ}) : Y → Y = (Y_{μ}, q_{μμ′}, M), where M = {μ_{0}}, q_{μμ′} = q_{μ_{0}μ_{0}} = 1_{Y_{μ_{0}}} = 1_{Y}, Y_{μ} = Y_{μ_{0}} = Y, q_{μ} = q_{μ_{0}} = 1_{Y}.

Let f : X → Y and g : Y → X be constant uniform maps. Let (f_{μ}, φ) : X → Y and (g_{λ}, ψ) : Y → X be morphisms induced by f and g, respectively. By Lemma 3 we have

(f_{μ}, φ) · (g_{λ}, ψ) ∼ (1_{Y_{μ}}, 1_{M}), (g_{λ}, ψ) · (f_{μ}, φ) ∼ (1_{X_{λ}}, 1_{Λ}).

Let F : X → Y and G : Y → X be uniform shape morphisms with representatives f = [(f_{μ}, φ)] : X → Y and g = [(g_{λ}, ψ)] : Y → X. It is clear that F · G = I_{Y} and G · F = I_{X}, where I_{X} and I_{Y} are the identity uniform shape morphisms. Hence ush(X) = ush(Y), i.e. ush(X) = 0.

Now we will prove the second assertion. Since X is finitistic, there is an antisymmetric cofinite ANRU-resolution p = (p_{λ}) : X → X = (X_{λ}, p_{λλ′}, Λ) of X [14]. Using this ANRU-resolution, the construction in the argument (iii)⇒(i) of [5] and our Lemma 3 in the place of [5, Lemma 2] and also Lemma 4 in next, we can define the ARU-system Z = (Z_{λ}, q_{λλ′}, Λ^{∗}) and the morphism q = (q_{λ}) : X → Z.

Following the argument in [5] that the obtained resolution satisfies properties(R1) and (R2), we can easily show that q has properties (UR1) and (UR2). □

Remark. Following the approach by [9], we can construct the coherent semiuniform homotopy category and, hence, uniform strong shape theory for uniform spaces by the inverse system approach based on semi-uniform homotopy. It is possible to find a condition in terms of uniform strong shape for a uniform finitistic space to have an ARU-resolution.

3. Uniform Shape Retracts

In this section we generalize the notion of an absolute shape retract [7] to the uniform spaces. Let A be a subspace of uniform space X. A uniform shape retraction is a uniform shape morphism R : X → A such that R · J = I_{A}, where J : A → X is the uniform shape morphism induced by the inclusion uniform map j : A → X. In this case the subspace A is called a uniform shape retract of X. Notice that if a uniform shape morphism R : X → A is induced by a uniform retraction r : X → A, then R is a uniform shape retraction. If A ⊆ Y ⊆ X and A is a uniform shape retract of X, then A is also a uniform shape retract of Y. Moreover, if A is a uniform shape retract of Y and Y is a uniform shape retract of X, then A is a uniform shape retract of X.

A uniform space X is called an absolute uniform shape retract provided for every uniform space Y with X ⊂ Y, X is a uniform shape retract of Y.

We have

Theorem 5. Let X be a uniform space. Then the following assertions are equivalent:

(i) X is an absolute uniform shape retract.

(ii) X has trivial uniform shape.

We first establish

Lemma 6. Let X and Y be uniform spaces. If Y has trivial uniform shape, then there exists a unique uniform shape morphism F : X → Y.

Proof. Let ush(Y) = 0. The uniform shape morphism C : Y → {∗}, induced by the constant map c : Y → {∗}, has an inverse uniform shape morphism G : {∗} → Y, G · C = I_{Y}. For each uniform shape morphism F : X → Y we have F = G · C · F = G · C′, where C′ = C · F : X → {∗} is the only uniform shape morphism into {∗}. □

Proof of Theorem 5. Implication (i)⇒(ii). Let X be an absolute uniform shape retract. We can consider X as a uniform subspace of an ARU-space Z. Note that Z is a semi-uniform contractible space. Consequently, Z has the semiuniform homotopy type of a point, and ush(Z) = 0. By condition, there exists a uniform shape retraction R : Z → X, R · J = I_{X}, where J : X → Z is the uniform shape morphism induced by the inclusion uniform map j : X → Z. From Lemma 6 it follows that J · R = I_{Z}. Thus ush(X) = ush(Z) = 0.

Implication (ii)⇒(i). Let ush(X) = 0 and X ⊂ Y. By Lemma 6 there exist unique uniform shape morphisms R : Y → X and I_{X} : X → X. Therefore R · J = I_{X}, i.e. X is an absolute uniform shape retract. □

Corollary 7. Let F : A → Y be a uniform shape morphism from a uniform subspace A of a uniform space X into an absolute uniform shape retract Y. Then there exists a uniform shape morphism F′ : X → Y such that F′ · J = F, where J : A → X is the inclusion uniform shape morphism induced by the inclusion uniform map j : A → X.

Proof. From Theorem 6 we obtain ush(Y) = 0. Then by Lemma 6 there are unique shape morphisms F : A → Y and F′ : X → Y. Consequently, the uniform shape morphism F′ · J : A → Y coincides with F : A → Y. □

Consider a uniform space X as a uniform subspace of an ARU-space Z. The set Λ = {U} of all uniform neighborhoods of X in Z is directed by the order ≤ defined by inclusion:

U ≤ U′ ⇔ U′ ⊆ U.

Let i_{U} : X → U and i_{UU′} : U′ → U be the inclusion uniform maps. The family i = ([i_{U}]) : X → X = (U, [i_{UU′}], Λ) forms a morphism of category pro-H(ANRU). In [11] it is proved that i : X → X is an H(ANRU)-expansion.

Theorem 8. A uniform subspace X of an ARU-space Z has trivial uniform shape if and only if for every uniform neighborhood U of X in Z there exists a uniform neighborhood U′ of X in Z such that U′ ⊆ U and the inclusion uniform map i_{UU′} : U′ → U is semi-uniformly homotopic to a constant map.

Proof. We can prove this as in S. Mardešic [7, Theorem 5], using Lemma 7. □

Theorem 9. For every finitistic uniform space X the following statements are equivalent:

(i) X admits an ARU-resolution.

(ii) X has trivial uniform shape.

(iii) X is an absolute uniform shape retract.

Proof. This is an immediate consequence of Theorems 1 and 5. □

Acknowledgement

The author expresses his gratitude to the referee for his helpful remarks.

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(Received 21.06.2002)

Author’s addresses:

Department of Mechanics and Mathematics I. Javakhishvili Tbilisi State University 2, University St., Tbilisi 0143 Georgia

Department of Physics and Mathematics Batumi State University 35, Ninoshvili St., Batumi 6016 Georgia E-mail: vbaladze@viam.hepi.edu.ge