Internal Neighbourhood Spaces

TL;DR

This paper characterizes internal neighbourhood operators in certain categories, establishing their equivalences, topological nature, and connections to bitopological spaces and locales, thus advancing the categorical understanding of topology.

Contribution

It provides a categorical framework for internal neighbourhoods, proving their equivalences and topological properties, and links them to known structures like bitopological spaces and locales.

Findings

Categories of internal neighbourhoods are topological.

Equivalence between neighbourhoods, Kuratowski operators, and pseudo-frame sets.

Internal neighbourhoods in topological spaces correspond to bitopological spaces.

Abstract

The main aim of this paper is to provide a description of neighbourhood operators in finitely complete categories with finite coproducts and a proper factorisation system such that the semilattice of admissible subobjects make a distributive complete lattice. The equivalence between neighbourhoods, Kuratowski interior operators and pseudo-frame sets is proved. Furthermore the categories of internal neighbourhoods is shown to be topological. Regular epimorphisms of categories of neighbourhoods are described and conditions ensuring hereditary regular epimorphisms are probed. It is shown the category of internal neighbourhoods of topological spaces is the category of bitopological spaces, while in the category of locales every locale comes equipped with a natural internal topology.