Mutually Compactificable Topological Spaces

Abstract

Two disjoint topological spacesX,Yare(T2-)mutually compactificable if there exists a compact(T2-)topology onK=X∪Ywhich coincides onX,Ywith their original topologies such that the pointsx∈X,y∈Yhave open disjoint neighborhoods inK. This paper, the first one from a series, contains some initial investigations of the notion. Some key properties are the following: a topological space is mutually compactificable with some space if and only if it isθ-regular. A regular space on which every real-valued continuous function is constant is mutually compactificable with noS2-space. On the other hand, there exists a regular non-T3.5space which is mutually compactificable with the infinite countable discrete space.