Revista Colombiana de Matemáticas Volumen 31 (1997), páginas 109–114

On the co-completeness of the category of Hausdorff uniform spaces

Serafín Bautista Januario Varela

Universidad Nacional de Colombia, Bogotá

Abstract. A construction of colimits in the category of Hausdorff uniform spaces is carried out by means of a family of pseudometrics. Since this is not a topological category, its co-completeness can not be ensured by the standard procedures. A careful revision of the usual arguments is then required to this end. Key words and phrases. Hausdorff uniform spaces, colimits, co-completeness. 1991 Mathematics Subject Classification. Primary 18A30. Secondary 18B99.

1. Uniformities in terms of pseudometrics

For many purposes, the appropriate generalization of the concept of a metric space is that of a Hausdorff uniform space, and not merely that of a uniform space. The latter could rather be viewed as a generalization of the notion of a pseudometric space. We focus our attention in this paper on the category Unif_{H} of Hausdorff uniform spaces and uniformly continuous maps, to establish the existence of colimits. Although all topological categories are known to be co-complete, and the category Unif of uniform spaces and uniformly continuous maps is such while Unif_{H} is not, the construction of colimits in this latter category deviates somewhat from the standard procedures followed for the former. First we examine some basic and preliminary facts.

A collection D of pseudometrics on a set X (with values in the real line ℝ) defines a uniformity on X by means of the subbasis of entourages {V_{d}^{ε} : d ∈ D, ε > 0}, where V_{d}^{ε} = {(x, y) ∈ X × X : d(x, y) < ε}. This uniformity is known to be the coarsest on X such that each d ∈ D is uniformly continuous for the product uniformity on X × X and the usual additive uniformity on ℝ.

1.1. Definition. Let (X, U) be a uniform space. The collection C_{X} of all finite uniformly continuous pseudometrics on X × X is called the total caliber of (X, U).

The following two well known results can be found in various textbooks on General Topology. For instance, in [6] pages 183 and 188.

1.2. Theorem. Let (X, U) be a uniform space and d: X × X ⟶ ℝ be a finite pseudometric. Then d is uniformly continuous if and only if V_{d}^{ε} ∈ U, for every ε > 0.

1.3. Theorem. Every uniformity on X is generated by its total caliber.

The next preliminary result is due to R. De Castro, see [4].

1.4. Theorem. Let (X, U), (Y, V) be uniform spaces with respective total calibers C_{X} and C_{Y}, and let f: X ⟶ Y. The following assertions are equivalent:

(i) f is uniformly continuous.

(ii) There exists a unique function ℓ: C_{Y} ⟶ C_{X}, d ⟼ d_{ℓ}, such that

d(f(x), f(y)) = d_{ℓ}(x, y) for all x, y ∈ X. (iii) There exists a function k: C_{Y} ⟶ C_{X}, d ⟼ d_{k}, such that

d(f(x), f(y)) ≤ d_{k}(x, y) for all x, y ∈ X.

Proof. (i) ⟹ (ii) Assume that f is uniformly continuous and let d be a pseudometric in C_{Y}. Then the map d′: X × X ⟶ ℝ defined by (x, y) ⟼ d′(x, y) = d(f(x), f(y)) is readily seen to be a pseudometric, and since f is uniformly continuous, for a given V_{d}^{ε} ∈ V there exists U ∈ U such that (x, y) ∈ U implies (f(x), f(y)) ∈ V_{d}^{ε}. Thus U ⊆ V_{d′}^{ε}, and then V_{d′}^{ε} ∈ U. Theorem 1.2 then implies that d′ is uniformly continuous, and consequently an element of the total caliber C_{X}. The conclusion in item (ii) is then obtained by taking d_{ℓ} = d′ for each d ∈ C_{Y}.

(ii) ⟹ (iii) If ℓ is as specified in (ii), the choice k = ℓ also satisfies (iii).

(iii) ⟹ (i) Assume k: C_{Y} ⟶ C_{X} is as specified in item (iii) in the statement of the theorem. Given V_{d}^{ε} ∈ V, take V_{d_{k}}^{ε} ∈ U. Since (x, y) ∈ V_{d_{k}}^{ε} implies (f(x), f(y)) ∈ V_{d}^{ε}, then f is uniformly continuous. □ ✓

2. A construction

2.1. Definition. Given a uniform space (Z, W), an arbitrary set X and a surjective map h: Z ⟶ X, the quotient uniformity on X by h is the uniformity {U ⊆ X × X : (h × h)^{−1}(U) ∈ W}.

2.2. Definition. Let (X_{i})_{i∈I} be a family of disjoint uniform spaces, and for each i ∈ I, call U_{i} the uniformity of X_{i}. The sum uniformity on Z = ⋃_{i∈I} X_{i} is the uniformity {V ⊆ Z × Z : for each i ∈ I, V ∩ (X_{i} × X_{i}) ∈ U_{i}}.

2.3. Remark. As a point of reference and comparison, recall that the colimit of the direct system ((X_{α}, U_{α})_{α∈Λ}, (f_{αβ})_{(α,β)∈Λ_{1}}) in Unif, where Λ is a directed set and Λ_{1} is the set {(α, β) ∈ Λ × Λ: α ≤ β}, is defined in a similar manner as to that in the category Set. That is, as the uniform space (Z/R, U), where Z is the disjoint union of the spaces X_{α} equipped with the sum uniformity, R is the equivalence relation given by x_{α} R x_{β} ⟺ ∃γ ≥ α, β such that f_{αγ}(x_{α}) = f_{βγ}(x_{β}), and U is the quotient uniformity on Z/R by the canonical map ϕ : Z ⟶ Z/R, x_{α} ⟼ \overline{x}_{α}. Nevertheless, in the case of the category of Hausdorff uniform spaces the construction has to be modified somewhat, as shown below.

Denote by Unif_{H} the category of Hausdorff uniform spaces and uniformly continuous maps, and by ((X_{α}, U_{α})_{α∈Λ}, (f_{αβ})_{(α,β)∈Λ_{1}}), a direct system in Unif_{H}. For each α ∈ Λ, let C_{α} denote the total caliber of the uniform space (X_{α}, U_{α}), and for a given uniformly continuous function f_{αβ}: X_{α} ⟶ X_{β}, let the unique map ℓ: C_{β} ⟶ C_{α} such that d_{β}(f_{αβ}(x_{α}), f_{αβ}(y_{α})) = (d_{β})_{ℓ}(x_{α}, y_{α}) be denoted by ℓ_{αβ}.

2.4. Theorem. The direct system ((X_{α}, U_{α})_{α∈Λ}, (f_{αβ})_{(α,β)∈Λ_{1}}) in Unif_{H} has a colimit.

Proof. The construction is carried on in the next four steps.

(1) Let Z = ⋃^{.}_{α∈Λ}X_{α} be the disjoint union of the family (X_{α})_{α∈Λ} and let C = \overleftarrow{lim} C_{α} be the projective limit of the system C_{α} with maps ℓ_{αβ}; that is, let

C = {d = (d_{α})_{α∈Λ} ∈ ∏_{α∈Λ} C_{α}: (d_{β})_{ℓ_{αβ}} = d_{α} for each (α, β) ∈ Λ_{1}}.

Consider on Z a collection of subbasic entourages B = {W_{d}^{ε} : ε > 0, d ∈ C}, where

W_{d}^{ε} = {(x_{α}, y_{β}) ∈ Z ×Z : ∃γ ∈ Λ s. t. γ ≥ α, β & (f_{αγ}(x_{α}), f_{βγ}(y_{β})) ∈ V_{d_{γ}}^{ε} }.

For each ε > 0 and d ∈ C we have △_{Z} ⊆ W_{d}^{ε}, W_{d}^{ε/2} ∘ W_{d}^{ε/2} ⊆ W_{d}^{ε} and (W_{d}^{ε})^{−1} = W_{d}^{ε}. Hence, B is a subbasis for a uniformity W on Z such that all the canonical injections j_{α}: X_{α} ⟶ Z are uniformly continuous. Furthermore, the uniformity W on Z is coarser than the sum uniformity. In fact, V_{d_{α}}^{ε} ⊆ W_{d}^{ε} ∩ (X_{α} × X_{α}), and V_{d_{α}}^{ε} ∈ U_{α} for each α ∈ Λ.

(2) Define on Z the equivalence relation

x_{α} R y_{β} ⟺ (∀d ∈ C)(∀ε > 0)((x_{α}, y_{β}) ∈ W_{d}^{ε}).

(3) Let X = Z/R and ϕ: Z ⟶ X, x_{α} ⟼ \overline{x}_{α}, be the quotient map. The collection {(ϕ×ϕ) (W_{d}^{ε}) : W_{d}^{ε} ∈ B} is a subbasis of entourages for a uniformity U on X. It is apparent that the map ϕ is uniformly continuous with respect to the uniformities W and U above.

The uniformity U is in general strictly coarser than the quotient uniformity on X by ϕ. In fact, if W ∈ W then ⋂_{j∈J} W_{d_{j}}^{ε} ⊆ W for some collection in C indexed by a finite set J and some ε > 0. But (ϕ × ϕ)( ⋂_{j∈J} W_{d_{j}}^{ε}) ⊆ ⋂_{j∈J}(ϕ × ϕ)(W_{d_{j}}^{ε}), and it may well happen that these two sets are different. Then, it can not be guaranteed that (ϕ × ϕ)(W) contains a finite intersection of sets of the form (ϕ × ϕ)(W_{d}^{ε}).

We claim that (X, U) is a Hausdorff space. Suppose that (\overline{x}_{α}, \overline{y}_{β}) ∈ (ϕ × ϕ) (W_{d}^{ε/3}) for all ε > 0 and all d ∈ C. Now, since for each ε > 0 and each d ∈ C there exists (u_{θ}, v_{ξ}) ∈ W_{d}^{ε/3} such that \overline{x}_{α} = \overline{u}_{θ} , \overline{y}_{β} = \overline{v}_{ξ}, so that (x_{α}, u_{θ}) ∈ W_{d}^{ε/3} , (y_{β}, v_{ξ}) ∈ W_{d}^{ε/3}, then for every ε > 0 and every d ∈ C we have that (x_{α}, y_{β}) ∈ W_{d}^{ε}, and therefore \overline{x}_{α} = \overline{y}_{β}.

(4) (X, U) is the sought colimit. In fact, define for each α ∈ Λ, τ_{α}: X_{α} ⟶ X, x_{α} ⟼ \overline{x}_{α}. The map τ_{α} is a morphism in Unif_{H}, because τ_{α} = ϕ ∘ j_{α}. But since x_{α} R f_{αβ}(x_{α}), it follows that τ_{β} ∘ f_{αβ} = τ_{α} if α ≤ β.

Given a second inductive cone for the given system, say for instance ((Y, V), (σ_{α}: X_{α} ⟶ Y)_{α∈Λ}), we claim that ψ: X ⟶ Y, \overline{x}_{α} ⟼ σ_{α}(x_{α}), is a morphism in Unif_{H} (necessarily unique) such that for each α ∈ Λ, ψ ∘ τ_{α} = σ_{α}. Indeed:

(a) ψ is a well defined map. To see this, let C_{Y} be the total caliber of (Y, V). The uniform continuity of σ_{α} and Theorem 1.4 above imply that for each α ∈ Λ there exists q_{α}: C_{Y} ⟶ C_{α}, d ⟼ d_{q_{α}}, such that for each d ∈ C_{Y} and for each x_{α}, y_{α} ∈ X_{α}, d(σ_{α}(x_{α}), σ_{α}(y_{α})) = d_{q_{α}}(x_{α}, y_{α}). Hence, it follows that for each d ∈ C_{Y}, each (α, β) ∈ Λ_{1} and each (x_{α}, y_{α}) ∈ X_{α} × X_{α},

d_{q_{α}}(x_{α}, y_{α}) = d(σ_{α}(x_{α}), σ_{α}(y_{α})) = d(σ_{β}(f_{αβ}(x_{α})), σ_{β}(f_{αβ}(y_{α})))
= d_{q_{β}}(f_{αβ}(x_{α}), f_{αβ}(y_{α})) = (d_{q_{β}})_{ℓ_{αβ}}(x_{α}, y_{α}).

Thus, for every d ∈ C_{Y}, d_{q_{α}} = d_{ℓ_{αβ}∘q_{β}}; that is, q_{α} = ℓ_{αβ}∘q_{β} for each (α, β) ∈ Λ_{1}.

Taking into account that C = \overleftarrow{lim} C_{α}, there exists a unique map φ: C_{Y} ⟶ C, d ⟼ φ(d) = (d_{q_{α}})_{α∈Λ}, such that, for each α ∈ Λ, π_{α} ∘ φ = q_{α}.

Assume that \overline{x}_{α} = \overline{y}_{β}. Given d ∈ C_{Y}, take d = φ(d). Then, for every ε > 0 there exists γ ∈ Λ with γ ≥ α, β such that (f_{αγ}(x_{α}), f_{βγ}(y_{β})) ∈ V_{d_{qγ}}^{ε}. Hence,

(∀d ∈ C_{Y})(∀ε > 0)( (σ_{α}(x_{α}), σ_{β}(y_{β})) = (σ_{γ} × σ_{γ})(f_{αγ}(x_{α}), f_{βγ}(y_{β})) ∈ V_{d}^{ε}).

Now, since (Y, V) is Hausdorff and {V_{d}^{ε} : ε > 0 , d ∈ C_{Y} } is a basis of entourages of V, it follows that ψ(\overline{x}_{α}) = σ_{α}(x_{α}) = σ_{β}(y_{β}) = ψ(\overline{y}_{β}). Thus, ψ is a well defined map, as claimed.

(b) ψ is a morphism of Unif_{H}. Let V_{d}^{ε} ∈ V. Then

(ψ × ψ)^{−1}(V_{d}^{ε})

= \{(\overline{x}_{α}, \overline{y}_{β}) : (ψ × ψ)(\overline{x}_{α}, \overline{y}_{β}) ∈ V_{d}^{ε}\}

= {(ϕ × ϕ)(x_{α}, y_{β}) : (σ_{α}(x_{α}), σ_{β}(y_{β})) ∈ V_{d}^{ε}}

= {(ϕ × ϕ)(x_{α}, y_{β}) : (∃γ ≥ α, β)[(σ_{γ}(f_{αγ}(x_{α})), σ_{γ}(f_{βγ}(y_{β}))) ∈ V_{d}^{ε}]}

= {(ϕ × ϕ)(x_{α}, y_{β}) ∈ X × X : (∃γ ≥ α, β)[(f_{αγ}(x_{α}), f_{βγ}(y_{β})) ∈ V_{q_{γ}(d)}^{ε}]}

= {ϕ × ϕ)(x_{α}, y_{β}) ∈ X × X : (x_{α}, y_{β}) ∈ W_{φ(d)}^{ε}} = (ϕ × ϕ)(W_{φ(d)}^{ε}) ∈ U. □ ✓

Unif_{H} has a closely related category in which the construction of colimits is straightforward. Let Pmet_{H} be the category whose objects are sets endowed with a family of pseudometrics (X, (d_{i})_{i∈I}) such that x = y if d_{i}(x, y) = 0 for each i ∈ I, and, given a second object (Y, (d_{j})_{j∈J}), a morphism is a pair of functions (f, k) : X × J ⟶ Y × I, (x, j) ⟼ (f(x), j_{k}), such that d_{j}(f(x), f(y)) ≤ d_{j_{k}}(x, y) for each x, y ∈ X.

2.5. Theorem. A direct system ((X_{α}, (d_{i_{α}})_{i_{α}∈I_{α}})_{α∈Λ}, (f_{αβ}, k_{αβ})_{(α,β)∈Λ_{1}}) in Pmet_{H} has a colimit

Proof. Let Z = ⋃^{.}_{α∈Λ}X_{α} be the disjoint union of the family (X_{α})_{α∈Λ}. Let I be the limit of the inverse system ((I_{α})_{α∈Λ}, (k_{αβ})_{(α,β)∈Λ_{1}}) in the category Set. For each i ∈ I consider the pseudometric δ_{i}: Z × Z ⟶ \overline{ℝ}, (x_{α}, y_{β}) ⟼ inf_{γ∈Λ} {d_{i_{γ}} (f_{αγ}(x_{α}), f_{βγ}(y_{β})): α, β ≤ γ}. Let R be the equivalence relation on Z defined by x_{α} R y_{β} if δ_{i}(x_{α}, y_{β}) = 0 for each i ∈ I.

If X = Z/R then \overline{δ}_{i}: X × X ⟶ \overline{ℝ}, (\overline{x}_{α}, \overline{y}_{β}) ⟼ δ_{i}(x_{α}, y_{β}) is a well defined pseudometric on X for each i ∈ I, as it can be easily verified. Consequently, (X, (\overline{δ}_{i})_{i∈I}) is an object of Pmet_{H}.

For each α ∈ Λ define τ_{α}: X_{α} ⟶ X by x_{α} ⟼ \overline{x}_{α}. Then (τ_{α}, π_{α}) is a morphism in Pmet_{H}, and ((X, (\overline{δ}_{i})_{i∈I}), ((τ_{α}, π_{α}): X_{α} × I ⟶ X × I_{α})_{α∈Λ}) is an inductive cone which is the colimit of the given direct system. The separation property of the objects in Pmet_{H} is necessary to establish the universal property of this cone. □ ✓

Let Calib be the category whose objects (X, C_{X}) are sets endowed with a family of pseudometrics coinciding with the total caliber of the underlying uniformity. Given a second object (Y, C_{Y}), a morphism is a pair of functions (f, ℓ) : X × C_{Y} ⟶ Y × C_{X}, (x, d) ⟼ (f(x), d_{ℓ}), such that d(f(x), f(y)) = d_{ℓ}(x, y) for each x, y ∈ X.

2.6. Proposition. The category Unif is equivalent to Calib.

Proof. Let F: Unif ⟶ Calib be the functor defined by F (X, U) = (X, C_{X}), where C_{X} is the total caliber of the uniformity U, and by F (f) = (f, ℓ): X ×C_{Y} ⟶ Y ×C_{X}, where ℓ is as in Theorem 1.4., if f: X ⟶ Y is a uniformly continuous map. Then F is readily seen to be full, faithful and isomorphism– dense. Hence, the asserted equivalence of categories (see [2]) follows. □ ✓

References

[1] Jir̆í Adámek et al., Abstract and Concrete Categories. The Joy of Cats, Wiley & Sons, 1989.

[2] Serafín Bautista, Co-completez de Categorías de Espacios Uniformes y Procesos de Lacalización, Tesis de Maestría, Programa de Posgrado en Matemáticas, Universidad Nacional, Bogotá, 1995.

[3] Nicolas Bourbaki, General Topology, Addison-Wesley, 1966.

[4] Rodrigo De Castro, Localización en Campos de Espacios Uniformes, Tesis de Maestría, Programa de Posgrado en Matemáticas, Universidad Nacional, Bogotá, 1985.

[5] Rodrigo De Castro & Januario Varela, Localization in bundles of uniform spaces, Rev. Colombiana Mat. 24 (1990), 103–112.

[6] John Kelly, General Topology, Van Nostrand, 1955.

[7] Saunders Mac Lane, Categories for the Working Mathematician, Springer-Verlag, 1971.

[8] Stephen Willard, General Topology, Addison-Wesley, 1970.

(Recibido en agosto de 1996; revisado en agosto de 1997)

Departamento de Matemáticas Universidad Nacional de Colombia

Bogotá, COLOMBIA

e-mail: sebadi@matematicas.unal.edu.co e-mail: jvarela@gaitana.interred.net.co