Mutually compactificable topological spaces (Q925362)

No separation axioms are assumed in topological spaces in this paper, which first discusses the notion of \(\theta\)-regularity of a topological space \(X\), i.e., every filter base of \(X\) with a \(\theta\)-cluster point has a cluster point. (A point \(x\in X\) is \(\theta\)-cluster point if \(x \in \cap \{\text{cl}_{\theta}F \mid F \in\) the filterbase\}, where \(x\) is in the \(\theta\)-closure of \(F\), \(\text{cl}_{\theta}F\), if every closed neighborhood of \(x\) intersects \(F\).) A space \(X\) is said to be compactificable by a space \(Y\) (or \(X\) and \(Y\) are mutually compactificable) if \(X \cap Y = \emptyset\) and there exists a compact topology on \(K = X \cup Y\) extending the topologies on \(X\) and \(Y\) such that two points in \(X\) and in \(Y\), respectively, are separated in \(K\), i.e., they have disjoint neighborhoods. A \(\theta\)-regular space is compactificable by its Wallman remainder, \(\omega X \setminus X\), which is also \(\theta\)-regular. The author shows that a necessary and sufficient condition for \(X\) to be compactificable by a space Y is that X is \(\theta\)-regular. \textit{A. Császár}'s separation axiom \(S_2\) [Introduction to general topology. Translated by Mrs. A. Csaszar. Bristol: Adam Hilger Ltd. Budapest: Akademiai Kiado (1978; Zbl 0366.54001)] is used: Two points with an open set containing only one of them necessarily have open disjoint neighborhoods. A space is regular (no \(T_1\) assumed) if and only if it is \(\theta\)-regular and \(S_2\). A regular space on which every continuous real-valued function is constant (hence not \(T_{3.5}\)) is mutually compactificable by no \(S_2\) space. An example of a regular non-\(T_{3.5}\) space is shown that is mutually compactificable with the infinite countable discrete space. Regular spaces do not form a closed class with respect to mutual compactificability. Any two disjoint strongly locally compact (i.e., every point in the space has a compact closed neighborhood) spaces are mutually compactificable. An example is shown in which a non-locally compact space is \(T_2\)-compactificable (i.e., \(K\) is Hausdorff) by a strongly locally compact space. In general, if \(X\) is compactificable by some compact space \(Y\), then \(X\) is strongly locally compact. This paper is the first in the author's program of investigating mutual compactificability.