THE FORGETFUL FUNCTOR Cont → Prox IS TOPOLOGICAL

Abstract

It is shown that the forgetful functor from the category of contiguity spaces to the category of generalized proximity spaces is topological, and that the right adjoint right inverse of this functor extends the inverse of the forgetful functor from the category of totally bounded uniform spaces to the category of proximity spaces.     

EXPONENTIAL LAW AND θ-CONTINUOUS FUNCTIONS

Abstract

The category θ-Top of topological spaces and θ-continuous functions is not Cartesian closed; but it is known that under certain local property assumptions, the exponential law in θ-Top is fulfilled. We define a functor from θ-Top to the category of H-θ-topological spaces and prove that in this category the exponential law holds without any local property assumptions. We also provide a functor from θ-Top to Katětov's category of filter-merotopic spaces, which is Cartesian closed.     

Idempotent probability measures on ultrametric spaces

Following the construction due to Hartog and Vink we introduce a metric on the set of idempotent probability measures (Maslov measures) defined on an ultrametric space. This construction determines a functor on the category of ultrametric spaces and nonexpanding maps. We prove that this functor is the functorial part of a monad on this category. This monad turns out to contain the hyperspace monad.     

TOPOLOGICAL CATEGORIES AND CLOSURE OPERATORS

Abstract

It is shown that the category CS of closure spaces is a topological category. For each epireflective subcategory A of a topological category X a functor F _A :X → X is defined and used to extend to the general case of topological categories some results given in [4], [5] and [10] for epireflective subcategories of the category Top of topological spaces.     

Intrinsic characterization of Alexander-Spanier cohomology groups of compactifications

In this paper we construct a uniform Alexander-Spanier cohomology functor from the category of pairs of uniform spaces to the category of abelian groups. We show that this functor satisfies all Eilenberg-Steenrod axioms on the category of pairs of precompact uniform spaces, is precompact uniform shape invariant and intrinsically, in terms of uniform structures, describes the Alexander-Spanier cohomology groups of compactifications of completely regular spaces.     

Strong inclusion orders between L-subsets and its applications in L-convex spaces

Abstract

In this paper, the concept of strong inclusion orders between L-subsets is introduced. As a tool, it is applied to the following aspects. Firstly, the notion of algebraic L-closure operators is proposed and the resulting category is shown to be isomorphic to the category of L-convex spaces (also called algebraic L-closure spaces). Secondly, restricted L-hull operators, as generalizations of restricted hull operators, are introduced and the resulting category is also proved to be isomorphic to the category of L-convex spaces. Finally, by using the properties of strong inclusion orders, it is shown that the category of convex spaces can be embedded in the category of stratified L-convex spaces as a reflective subcategory and the concrete form of the coreflective functor from the category of L-convex spaces to the category of stratified L-convex spaces is presented.     

Dimension zero at all scales

We consider the notion of dimension in four categories: the category of (unbounded) separable metric spaces and (metrically proper) Lipschitz maps, and the category of (unbounded) separable metric spaces and (metrically proper) uniform maps. A unified treatment is given to the large scale dimension and the small scale dimension. We show that in all categories a space has dimension zero if and only if it is equivalent to an ultrametric space. Also, 0-dimensional spaces are characterized by means of retractions to subspaces. There is a universal zero-dimensional space in all categories. In the Lipschitz Category spaces of dimension zero are characterized by means of extensions of maps to the unit 0-sphere. Any countable group of asymptotic dimension zero is coarsely equivalent to a direct sum of cyclic groups. We construct uncountably many examples of coarsely inequivalent ultrametric spaces.     

PROXIMAL LIMIT SPACES

Abstract

The proximal limit spaces are introduced which fill the gap arising from the existence of proximity spaces, uniform spaces, and uniform limit spaces. It is shown that the proximal limit spaces can be considered as a bireflective subcategory of the topological category of uniform limit spaces. A limit space is induced by a proximal limit space if and only if it is a S₁-limit space.     

The Cartesian product of a compactum and a space is a bifunctor in shape

In 2003 the author has associated with every cofinite inverse system of compact Hausdorff spaces X with limit X and every simplicial complex K (possibly infinite) with geometric realization P=|K| a resolution R(X,K) of X×P, which consists of paracompact spaces. If X consists of compact polyhedra, then R(X,K) consists of spaces having the homotopy type of polyhedra. In two subsequent papers the author proved that R(X,K) is a covariant functor in each of its variables X and K. In the present paper it is proved that R(X,K) is a bifunctor. Using this result, it is proved that the Cartesian product X×Z of a compact Hausdorff space X and a topological space Z is a bifunctor SSh(Cpt)×Sh(Top)→Sh(Top) from the product category of the strong shape category of compact Hausdorff spaces SSh(Cpt) and the shape category Sh(Top) of topological spaces to the category Sh(Top). This holds in spite of the fact that X×Z need not be a direct product in Sh(Top).     

Functoriality of the standard resolution of the Cartesian product of a compactum and a polyhedron

In a previous paper the author has associated with every inverse system of compact Hausdorff spaces X with limit X and every simplicial complex K (possibly infinite) with geometric realization P=|K| a resolution RK(X) of X×P, which consists of paracompact spaces. If X consists of compact polyhedra, then RK(X) consists of spaces having the homotopy type of polyhedra. In the present paper it is proved that this construction is functorial. One of the consequences is the existence of a functor from the strong shape category of compact Hausdorff spaces X to the shape category of spaces, which maps X to the Cartesian product X×P. Another consequence is the theorem which asserts that, for compact Hausdorff spaces X, X^′, such that X is strong shape dominated by X^′ and the Cartesian product X^′×P is a direct product in Sh(Top), then also X×P is a direct product in the shape category Sh(Top).     

Observations on quasi-uniform products

We prove that any product of quotient maps in the category of quasi-uniform spaces and quasi-uniformly continuous maps is a quotient map. We also show that a quasi-uniformly continuous map from a product of quasi-uniform spaces into a quasi-pseudometric T₀-space depends on countably many coordinates.Furthermore we characterize those quasi-uniformities that are unique in their quasi-proximity class and prove that this property is preserved under arbitrary products in the category of quasi-uniform spaces.     

On the MacNeille Completion of the Category of Partially Ordered Topological Spaces

Preordered topological spaces for which the order has a closed graph form a topological category. Within this category we identify the MacNeille completions (coinciding with the universal initial completions) of five monotopological subcategories, namely those of the T₀(T₁, T₂) preordered spaces and the (completely regular) partially ordered spaces. We also show that a functor due to L. NACHBIN from the quasi-uniform spaces to the preordered spaces preserves initial sources.     

Cartesian closed hull of the category of uniform spaces

A concrete category $\text{K}$ is a CCT (cartesian closed topological) extension of the category Unif of uniform spaces if 1. $\text{K}$ is cartesian closed, 2. Unif is a full, finitely productive subcategory of $\text{K}$ and the forgetful functor of $\text{K}$ extends that of Unif and 3. $\text{K}$ has initial structures. We describe the smallest CCT extension of Unif which is called the CCT hull by H. Herrlich and L.D. Nel. The objects of the CCT hull are bornological uniform spaces, i.e. uniform spaces endowed with a collection of “bounded” sets related naturally to the uniformity; the morphisms are the uniformly continuous maps which preserve the bounded sets.     

Filter spaces

The category FIL of filter spaces and cauchy maps is a topological universe. This paper establishes the foundation for a completion theory forT ₂ filter spaces.     

Cochains and homotopy type

Finite type nilpotent spaces are weakly equivalent if and only if their singular cochains are quasi-isomorphic as E_∞ algebras. The cochain functor from the homotopy category of finite type nilpotent spaces to the homotopy category of E_∞ algebras is faithful but not full.     

Injective Hulls in POSV

We determine the injective objects and hulls in the category POS_V. This category is similar to the one of join semilattices but contains all partially ordered sets. The results of this paper have applications, for instance, in the theory of (generalized) ultrametric spaces.     

Epireflective subcategories of valuated groups

We show that the category of valuated groups has a topological forgetful functor to the category of abelian groups. This category is universal, that is, it is the bireflective hull of its T_o-objects, and properties of the (large) lattice of epireflective subcategories are contrasted with results obtained by T. Marny [7] for universal categories over the category of sets.     

Completion of merotopic spaces and extension of uniformly continuous maps

A general Riesz merotopic space (X, ν) determines a not necessarily topological closure operator c_ν on X. The space (X, ν) is said to be complete if every cluster on (X, ν) is contained in an adherence grill on (X, c_ν). We discuss a method of obtaining a large class of completions of a given Riesz merotopic space with induced T₁ closure space. As special cases we get the simple completion, which induces a simple closure space extension, and the strict completion, which induces a strict closure space extension. We show that the category of complete separated T₁ Riesz merotopic spaces is epireflective in the category of separated T₁ Riesz merotopic spaces, the reflection of an object being the simple completion. Similarly the category of complete clan-covered quasi-regular T₁ Riesz merotopic spaces is epireflective in the category of clan-covered quasi-regular T₁ Riesz merotopic spaces, the reflection of an object being the strict completion.