United sight to an algebraic operations and convergence

TL;DR

This paper explores a generalized framework for algebraic operations using convergence spaces, introducing new notions of continuity and local properties that extend classical topological concepts.

Contribution

It introduces a novel approach to algebraic operations via convergence spaces, defining new types of continuity and local properties in this generalized setting.

Findings

Continuity defined via commuting squares in convergence spaces.

Introduction of adherence space as a special convergence.

Local continuity properties differ from classical global continuity.

Abstract

Algebraic operations are understood as topologiztion of algebra. They become an example of simplest convergence space. In our article the convergence is a arbitrary multivalued appointment. The continuity of some mapping between two convergence spaces is defined as a property of commuting squares. The adherence space is defined as a special convergence. The continuity in such spaces is understood as a local property, in contrary to the global continuity property in topological spaces. Possible bounded mappings in bornological spaces is also introduced, without deeper investigation. The local counterpart of bornological space is defined as proximity space. The new notions of continuity are non trivially also for a functional mappings, therefore they got their own names in English.