Mutually Compactificable Topological Spaces

Two disjoint topological spaces X, Y are (T2-) mutually compactificable if there exists a compact (T2-) topology on K=X∪Y which coincides on X, Y with their original topologies such that the points x∈X, y∈Y have open disjoint neighborhoods in K. 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 is θ-regular. A regular space on which every real-valued continuous function is constant is mutually compactificable with no S2-space. On the other hand, there exists a regular non-T3.5 space which is mutually compactificable with the infinite countable discrete space

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 X1, X2 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 T1 representative, but there are classes with no Hausdorff representatives