Category Of Uniform Spaces Research Articles

Relating Bishopʼs function spaces to neighbourhood spaces

We extend Bishopʼs concept of function spaces to the concept of pre-function spaces. We show that there is an adjunction between the category of neighbourhood spaces and the category of Φ-closed pre-function spaces. We also show that there is an adjunction between the category of uniform spaces and the category of Ψ-closed pre-function spaces.

Two subcategories of apartness spaces

We introduce the notion of a topological quasi-apartness space and the notion of a uniform quasi-apartness space, and construct an adjunction between the category of topological quasi-apartness spaces and the category of neighbourhood spaces, and an adjunction between the category of uniform spaces and the category of uniform quasi-apartness spaces.

Topological structure of direct limits in the category of uniform spaces

Let ( X n ) n ∈ ω be a sequence of uniform spaces such that each space X n is a subspace in X n + 1 . We give an explicit description of the topology and uniformity of the direct limit u - lim → X n of the sequence ( X n ) in the category of uniform spaces. This description implies that a function f : u - lim → X n → Y to a uniform space Y is continuous if for every n ∈ N the restriction f | X n is continuous and regular at the subset X n − 1 in the sense that for any entourages U ∈ U Y and V ∈ U X there is an entourage W ∈ U X such that for each point x ∈ B ( X n − 1 , W ) there is a point x ′ ∈ X n − 1 with ( x , x ′ ) ∈ V and ( f ( x ) , f ( x ′ ) ) ∈ U . Also we shall compare topologies of direct limits in various categories.

Uniform universal covers of uniform spaces

We develop a generalized covering space theory for a class of uniform spaces called coverable spaces. Coverable spaces include all geodesic metric spaces, connected and locally pathwise connected compact topological spaces, in particular Peano continua, as well as more pathological spaces like the topologist's sine curve. The uniform universal cover of a coverable space is a kind of generalized cover with universal and lifting properties in the category of uniform spaces and uniformly continuous mappings. Associated with the uniform universal cover is a functorial uniform space invariant called the deck group, which is related to the classical fundamental group by a natural homomorphism. We obtain some specific results for one-dimensional spaces.

Universality of Coproducts in Categories of Lax Algebras

Categories of lax \((T,V\,)\)-algebras are shown to have pullback-stable coproducts if \(T\) preserves inverse images. The general result not only gives a common proof of this property in many topological categories but also shows that important topological categories, like the category of uniform spaces, are not presentable as a category of lax \((T,V\,)\)-algebras, with \(T\) preserving inverse images. Moreover, we show that any such category of \((T,V\,)\)-algebras has a concrete, coproduct preserving functor into the category of topological spaces.

FuzzyTL-uniform spaces

The main purpose of this paper is to introduce a new structure that is a fuzzyTL-uniform space. We show that our structure generates a fuzzy topological space, precisely, a fuzzyT-locality space. Also, we deduce the concept of level uniformities of a fuzzyTL-uniformity. We connect the category of fuzzyTL-uniform spaces with the category of uniform spaces. We establish a necessary and sufficient condition, under which a fuzzyTL-uniformity is probabilistic pseudometrizable. Finally, we define a functor from the category of fuzzyTL-uniform spaces into the category of fuzzyT-locality spaces and we show that it preserves optimal lifts.

Universal categories of uniform and metric locales

Recall that a category is called universal if it contains an isomorphic copy of any concrete category as a full subcategory. In particular, if\(C\) is universal then every monoid can be represented as the endomorphism monoid of an object in\(C\). A major obstacle to universality in categories of topological nature are the constant maps (which prevent, for instance, representing nontrivial groups as endomorphism monoids). Thus, to obtain, say, a universal category of uniform spaces, the constants have to be prohibited by artificial additional conditions (for instance, conditions of an openness type). Since in generalized spaces (locales) we do not necessarily have points, the question naturally arises as to whether we can get rid of surplus conditions in search of universality there. In this paper we prove that the category of uniform locales with all uniform morphisms is universal. Indeed we establish the universality already for the subcategory of very special uniform locales, namely Boolean metric ones. Moreover, universality is also obtained for more general morphisms, such as Cauchy morphisms, as well as for special metric choices of morphisms (contractive, Lipschitz). The question whether one can avoid uniformities remains in general open: we do not know whether the category of all locales with all localic morphisms is universal. However, the answer is final for the Boolean case: by a result of McKenzie and Monk ([10], see Section 4) one cannot represent groups by endomorphisms of Boolean algebras without restriction by an additional structure.

THE IMPACT OF CLOSURE OPERATORS ON THE STRUCTURE OF A CONCRETE CATEGORY

Abstract We study the question of whether morphisms of a concrete category X can be determined by means of closure operators of X in the sense of [DT]. In particular, we show that closure operators cannot detect uniformly continuous maps in case X is the category of uniform spaces. We have analogous result for proximity spaces and topological modules.

How Topological are Topological Groups?

Abstract A Characterization of the category of topological groups is provided which does not refer to the category of topological spaces at all, but only to the category of uniform spaces. Similarly, the category of SIN-groups is characterized in a purely uniform way.

DISCRETE UNIFORM STRUCTURES AND QUASITOPOI

It is known that no non-trivial subcategory of the category of topological spares is a quasitopos in the sense of Penon. The purpose of this note is to establish the existence of a proper class of sub-categories of the category of uniform spaces which are quasitopoi. These subcategories are generated by certain proximally discrete uniform spaces which correspond to each infinite regular cardinal.

Fuzzy uniform spaces

In this paper we introduce the notion of fuzzy uniform spaces. In Section 2 we give some basic results and show how a fuzzy topology is derived from a fuzzy uniformity. In Sections 3 and 4 we prove that the notions which are introduced are good extensions and we prove that the category of uniform spaces is nicely injected in the category of fuzzy uniform spaces. Finally in Section 5 we prove some basic results

Coreflectors not preserving the interval and Baire partitions of uniform spaces

Values of the compact interval and other spaces under coreflectors in the category of uniform spaces are studied. It is shown that any coreflector which changes the usual uniformity of the interval produces a new uniformity which contains all finite Baire partitions of the interval.

The isomorphisms of the category of uniform spaces and related categories

In this paper we determine the isomorphisms of the categories of uniform spaces, proximity spaces and all full embeddings of the categories of several classes of topological spaces (corresponding to usual separation axioms) into themselves. In the lemmas we investigate the relation of the forgetful functors to concrete categories and to functors between concrete categories, in a general setting. To finish we mention several unsolved problems. Uniform and proximity spaces are not assumed to be separated. Unif denotes the category of uniform spaces, HUnif denotes that of separated uniform spaces. THEOREM. Every isomorphism i of Unif onto itself is of the following form: for each uniform space X there is an isomorphism ix:X~i(X), and for each map f: X~ Y the following diagram commutes: X f.~Y ix iv i(X) ~(f), i(Y)

A note on metric-fine spaces

The coreflection into metric-fine spaces X X is explicitly described, and it is shown that metric-fine proximally fine spaces are just the spaces X X such that f : X → Y f:X \to Y is uniformly continuous whenever the pre-images under f f of zero sets are zero sets.

The Homology of Uniform Spaces

In the past thirty years, algebraic topologists have developed a great body of knowledge concerning the category of topological spaces. By contrast, corresponding problems in the category of uniform spaces have been barely touched. Lubkin [8] studied the notion of a covering space in the category of generalized uniform spaces, and suggested that much of algebraic topology could be profitably studied in this category. Deming [2] discussed the fundamental group of a generalized uniform space, and related it to the first Čech homology group. A slightly different version of Cech cohomology was defined by Kuzminov and Svedov in [7] and related to the dimension theory of uniform spaces.

A Note on Cardinal Reflections in the Category of Uniform Spaces

A Note on Cardinal Reflections in the Category of Uniform Spaces

A note on cardinal reflections in the category of uniform spaces

In [5, Chapter iii, Exercise 2] it is shown that the countable uniform covers of any uniform space form a uniformity, and it is asked whether the same thing happens for arbitrary infinite cardinals. Here a partial affirmative answer is given: when its uniformity has a basis of a-point-finite uniform covers ( a family of sets is said to be a-pointfinite iff it is a union of countably many subfamilies each of which is a point-finite family), the space admits arbitrary cardinal reflections. This means that a positive solution of [5, Research Problem B3] implies a positive solution of the above question. But the major interest of our result is that it furnishes the expected and most natural characterization of the uniformities defined in [1, ?3] by extended reticles: they are the cardinal reflections of the fine uniformity of the given space. From this, a very easy proof of the main results of [2 ] is derived, and it seems that almost all the work of G. Aquaro in that field may be considerably simplified at least in the technical apparatus (this is certainly true for [3]). Our terminology and notations are based on [5] (but we do not assume a priori the axiom T2 on uniform spaces).