Internal Neighbourhood Structures II: Closure and closed morphisms
Abstract
Internal preneighbourhood spaces inside any finitely complete category with finite coproducts and proper factorisation structure were first introduced in my earlier paper. This paper proposes a closure operation on internal preneighbourhood spaces and investigates closed morphisms and its close allies. Consequently it introduces analogues of several well known classes of topological spaces for preneighbourhood spaces. Some preliminary properties of these spaces are established in this paper. The results of this paper exhibit preneighbourhood systems are more general than closure operators and conveniently allows identifying properties of classes of morphisms independent of continuity of morphisms with respect to induced closure operators.
Cite
@article{arxiv.2004.06238,
title = {Internal Neighbourhood Structures II: Closure and closed morphisms},
author = {Partha Pratim Ghosh},
journal= {arXiv preprint arXiv:2004.06238},
year = {2022}
}
Comments
The previous version used a notion of closure which was trivial --- I am thankful to the anonymous referee of TAC for pointing this out to me. This required a completely new working of the closure operation. Hopefully this time it faithfully and conservatively extends the well known notions