The Classes of Mutual Compactificability

Two disjoint topological spaces X, Y are mutuallycompactificable if there exists a compact topology on K=X∪Y which coincides on X, Y with their original topologies suchthat the points x∈X, y∈Y have disjoint neighborhoods in K. The main problem under consideration is the following: whichspaces X, Y are so compatible such that they together can formthe compact space K? In this paper we define and study theclasses of spaces with the similar behavior with respect to themutual compactificability. Two spaces X1, X2 belong to thesame class if they can substitute each other in the aboveconstruction with any space Y. In this way we transform the mainproblem to the study of relations between the compactificabilityclasses. Some conspicuous classes of topological spaces arediscovered as the classes of mutual compactificability. Thestudied classes form a certain “scale of noncompactness” fortopological spaces. Every class of mutual compactificabilitycontains a T1 representative, but there are classes with noHausdorff representatives