Internal neighbourhood structures

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.

Internal Neighbourhood Structures III: Finite Sum of Subobjects

“…The notion of an internal preneighbourhood space was initiated in [7]. The notion of a closure operator on an internal preneighbourhood space derived from its internal preneighbourhood system was introduced and detailed in [8].…”

“…The notion of a closure operator on an internal preneighbourhood space derived from its internal preneighbourhood system was introduced and detailed in [8]. The present section lists all definitions and results necessary for this paper; for details see [7,8].…”

“…The notion of internal preneighbourhood spaces were initiated in [7]. The sole purpose of this paper is to introduce the notion of an internal Hausdorff space and the Hausdorff reflection of an internal preneighbourhood space.…”

“…Contexts are ubiquitous, the most generic example being of a small complete small cocomplete well-powered category A with E being the set of all epimorphisms and M being the set of all extremal monomorphisms; other examples are listed on page 4. Contexts were introduced in [7] as a tool to introduce the notion of an internal neighbourhood space; in [8] the notion of closure and closed morphisms were developed and characterisations obtained for the sum admissible or closed morphisms to be again admissible or closed. In §2 all the definitions and facts necessary for this paper are recalled from [7,8].…”

“…Contexts were introduced in [7] as a tool to introduce the notion of an internal neighbourhood space; in [8] the notion of closure and closed morphisms were developed and characterisations obtained for the sum admissible or closed morphisms to be again admissible or closed. In §2 all the definitions and facts necessary for this paper are recalled from [7,8]. (b) The first set of results of this paper appear from §3 where the notion of a proper morphism (see Definition 3.1) is described.…”