The Classes of Mutual Compactificability

Abstract

Two disjoint topological spacesX,Yare mutually compactificable if there exists a compact topology onK=X∪Ywhich coincides onX,Ywith their original topologies such that the pointsx∈X,y∈Yhave disjoint neighborhoods inK. The main problem under consideration is the following: which spacesX,Yare so compatible such that they together can form the compact spaceK? In this paper we define and study the classes of spaces with the similar behavior with respect to the mutual compactificability. Two spacesX1,X2belong to the same class if they can substitute each other in the above construction with any spaceY. In this way we transform the main problem to the study of relations between the compactificability classes. Some conspicuous classes of topological spaces are discovered as the classes of mutual compactificability. The studied classes form a certain “scale of noncompactness” for topological spaces. Every class of mutual compactificability contains aT1representative, but there are classes with no Hausdorff representatives.