Apartness spaces were introduced as a constructive counterpart to proximity spaces which, in turn, aimed to model the concept of nearness of sets in a metric or topological environment. In this paper we introduce apartness algebras and apartness frames intended to be abstract counterparts to the apartness spaces of Bridges and Vîta [3], and we prove a discrete duality for them.

Partial frames provide a rich context in which to do pointfree structured and unstructured topology. A small collection of axioms of an elementary nature allows one to do much traditional pointfree topology, both on the level of frames or locales, and that of uniform or metric frames. These axioms are sufficiently general to include as examples bounded distributive...

Starting with the intuitive concept o r nearness" as a binar,, relation, semi-proximity spaces (sp-spaces) are defined. The restrictions on semi-proximity spaces are weaker than on topological proximity spaces. Thus, semi-proximity spaces generalize classical topological spaces. Moreover, it is possible to describe all digital pictures used in computer vision and computer graphics as non-trivia...

This article gives a brief overview of near sets. The proposed approach in introducing near sets is to consider a set theory-based form of nearness (proximity) called discrete proximity. There are two basic types of near sets, namely, spatially near sets and descriptively near sets. By endowing a nonempty set with some form of a nearness (proximity) relation, we obtain a structured set called a...

This paper introduces the concept of an approach merotopological space and studies its categorytheoretic properties. Various topological categories are shown to be embedded into the category whose objects are approach merotopological spaces. The order structure of the family of all approachmerotopologies on a nonempty set is discussed. Employing the theory of bunches, bunch completion of an app...

Completeness for metric spaces is traditionally presented in terms of convergence of Cauchy sequences, and for uniform spaces in terms of Cauchy filters. Somewhat more abstractly, a uniform space is complete if and only if it is closed in every uniform space in which it is embedded, and so isomorphic to any space in which it is densely embedded. This is the approach to completeness used in the ...

Several features of subfitness are analyzed. Isbell’s Spatiality Theorem leads to the concept of T1-spatiality which is compared with the TD-spatiality. Subfitness is put into relation with other weak separation axioms (and with the existence of nearness), both in spaces and locales, and seeming discrepancies are explained. A new characteristic of subfitness by means of codensity is presented.

Abstract This article considers the problem of how to formulate a framework for the study of the nearness of tolerance rough sets (TRS). The solution to the problem stems from recent work on near sets and approach spaces as well as from the realisation that disjoint TRSs can be viewed in the context of approach merotopic spaces. A set of TRSs equipped with a distance function satisfying certain...

An axiomatic development of the theory of apartness and nearness of a point and a set is introduced as a framework for constructive topology. Various notions of continuity of mappings between apartness spaces are compared; the constructive independence of one of the axioms from the others is demonstrated; and the product apartness structure is defined and analysed.