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It is our aim in this note to take a closer look at some separation axioms in the construct AP of approach spaces and contractions. Whereas lower separation axioms seem to be qualitative, the higher ones seem to have a quantitative nature. Also some characterizations for the corresponding epireectors will be given.

This paper studies the stability properties of the concepts of local compactness introduced by the authors in 1998. We show that all of these concepts are stable for contractive, expansive images and for products.

This paper introduces some definitions of a concept of local compactness in approach spaces. The basic relationships between these concepts is studied, and measures of local compactness are defined.

In order to have a topological category that contains both pMET and Unif in a nice way (and therefore combines quantitative and uniform concepts), approach uniformities are introduced. Approach uniformities uniformize the so-called approach spaces, introduced in Lowen [8]. Different characterizations of approach uniformities are formulated. Some natural examples are presented, such as function spaces, hyperspaces, spaces of measures, and an example concerning theoretical computer science.

In this paper we introduce the notions of separability and Lindelöf in approach spaces and investigate their behaviour under products and subspaces.

In a fuzzy topology on a set X, the limit of a prefilter (i.e. a filter in the lattice [0,1]X) is calculated from the fuzzy closure. In this way convergence is derived from a fuzzy topology. In our paper we start with any rule lim which to any prefilter 𝔇 on X assigns, a function lim𝔇∈[0,1]X. We give necessary and sufficient conditions for the function 𝔇→lim𝔇 in order that it can be derived from a fuzzy topology.

We show that convergence spaces with continuous maps and metric spaces with contractions, can be viewed as entities of the same kind. Both can be characterized by a limit function λ which with each filter ℱ associates a map λℱ from the underlying set to the extended positive real line. Continuous maps and contractions can both be characterized as limit function preserving maps.

It is shown that the category of non-Archimedean metric spaces with 1-Lipschitz maps can be embedded as a coreflective non-bireflective subcategory in the category of fuzzy uniform spaces. Consequential characterizations of topological and uniform properties are derived.

In this paper we prove a cancellation law in the category (1) of partially ordered sets and (2) of $T_0$ spaces, the respective conditions on the common factor being that it have (1) an irreducible bottom element and (2) an irreducible bottom element in its specialization order. [ABSTRACT FROM AUTHOR]