The Classes of Mutual Compactificability

Two disjoint topological spaces X , Y are mutually compactificable if there exists a compact topology on K = X ∪ Y which coincides on X , Y with their original topologies such that the points x ∈ X , y ∈ Y have disjoint neighborhoods in K . The main problem under consideration is the following: which spaces X , Y are so compatible such that they together can form the compact space K ? In this paper we define and study the classes of spaces with the similar behavior with respect to the mutual compactificability. Two spaces X 1 , X 2 belong to the same class if they can substitute each other in the above construction with any space Y . 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 a T 1 representative, but there are classes with no Hausdorff representatives.