Internal Neighbourhood Structures III: Finite Sum of Subobjects

Abstract

This paper develops further on internal preneighbourhood spaces [see 7, 8] in presenting the notion of internal Hausdorff spaces and the Hausdorff reflection. The development depends on the notions of proper morphisms, separated morphisms and separated objects, which are also developed in this paper. The Hausdorff reflection is described in three different ways: firstly as the largest subobject of the binary product whose components are indistinguishable by any internal Hausdorff space valued preneighbourhood morphism, secondly as the smallest effective equivalence relation whose quotient is an internal Hausdorff space and thirdly in admissibly well powered categories by transfinite induction on quotients by the diagonal.