Generalized topological spaces with associating function

TL;DR

This paper introduces a new framework for generalized topological spaces where points outside open sets are associated with neighborhoods via a function, extending classical concepts of interior, closure, convergence, and continuity.

Contribution

It proposes a novel approach to generalized topologies by incorporating an associating function, allowing for the study of spaces with non-closed families and non-open whole spaces.

Findings

Defined F-interior and F-closure concepts

Analyzed convergence and continuity in the new setting

Explored properties of spaces with associated points

Abstract

Generalized topological spaces in the sense of Cs\'{a}sz\'{a}r have two main features which distinguish them from typical topologies. First, these families of subsets are not closed under intersections. Second, we allow for the possibility that the whole space is not open. Hence, some points of the universe may be beyond any open set. In this paper we assume that such points are associated with certain open neighbourhoods by means of a special function F. We study various properties of the structures obtained in this way. We introduce the notions of F-interior and F-closure and we discuss issues of convergence and continuity in this new setting.