Some remarks on anti-topological spaces

TL;DR

This paper provides a comprehensive analysis of anti-topological spaces, exploring their foundational properties, interpretations of classical topological concepts, and introducing new types of continuity and logical frameworks.

Contribution

It offers the first detailed examination of anti-topological spaces, clarifies their relation to other generalized spaces, and proposes novel notions of continuity and logical interpretation.

Findings

Analysis of basic definitions and implications

Introduction of two non-equivalent continuity types

Preliminary ideas on modal logic for anti-topological spaces

Abstract

This paper is devoted to a general presentation of anti-topological spaces. These structures have been initially proposed by \c{S}ahin, Karg{\i}n and M. Y\"{u}cel in 2021. We analyse their basic definition, showing some of its subtleties and implications. The framework thus obtained is used to investigate anti-topological interpretation of some basic topological notions. For example, we discuss the idea of interior and closure and we show some results on door spaces. Moreover, we introduce two non-equivalent types of continuity. Finally, we investigate the idea of density and nowhere density. Finally, we give some preliminary suggestions concerning the modal logic of anti-topological spaces. It is noteworthy that the paper contains some additional remarks on infra-topological and weak spaces. They may be considered as a clarification or correction of some earlier results present in literature.