The classes of mutual compactificability (Q925350)

No separation axioms are assumed in this paper. The author argues for this and traces a line of study in non-Hausdorff compactness. The second of a series of articles on mutual compactificability, this paper deals with the ``compactificability'' of topological spaces in the sense that their union forms a compact space in a specific way. This is done via a study of a relationship between compactificability classes by introducing an equivalence relation \(\sim\) : \(X \sim Z\) if for every non-empty space \(Y\) disjoint from \(X\) and from \(Z\), \(X\) is compactificable by \(Y\) iff \(Z\) is. The equivalence class \(\mathcal{C}(X)\) containing \(X\) is the compactificability class of \(X\), thus the author obtains the notion of compactificability classes with an order relation. The author describes the compactificability classes of spaces constructed from the real line, the Cantor or Tikhonov cubes, in a subsequent paper. Non-\(\theta\)-regular spaces form a compactificability class called improper because no element of that class is compactificable by any space. The other classes are called proper. The set of all compact spaces forms a class of compactificability and is a maximal element in the ordering among all proper classes of compactificability. These classes form a certain ``scale'', measuring the level of non-compactness for spaces. For strongly locally compact spaces, the class of all compact spaces is the greatest element. For a regular space \(X\) on which every continuous real-valued function is constant, the compactificability class \(\mathcal{C}(\omega X \setminus X)\) of the Wallman remainder of \(X\) contains no Hausdorff representative. Every proper compactificability class contains a \(\theta\)-regular \(T_1\) representative.