Some properties of subcompact spaces (Q2113416)

Readers must guess that ZFC is the set-theoretic framework of this article. A continuous bijection of a topological space \(X\) onto a topological space \(Y\) is called a condensation. A compact Hausdorff space \(X\) is called: (1) an \(a\)-space if, for each countable subset \(C\) of \(X\), there exists a condensation of \(X\setminus C\) onto a compact Hausdorff space; (2) a strict \(a\)-space if, for each countable subset \(C\) of \(X\), there exists a condensation of \(X\setminus C\) onto a compact Hausdorff space \(Y\) which can be extended to a continuous mapping of \(X\) onto \(Y\). A Hausdorff space \(X\) is called: (3) subcompact if there exists a condensation of \(X\) onto a compact Hausdorff space; (4) \(a\)-subcompact if, for each countable subset \(C\) of \(X\), there exists a condensation of \(X\setminus C\) onto an \(a\)-space; (5) strictly \(a\)-subcompact if, for each countable subset \(C\) of \(X\), there exists a condensation of \(X\setminus C\) onto a strict \(a\)-space \(Y\) which has an extension to a continuous mapping of \(X\) onto \(Y\); (6) weakly \(a\)-subcompact if, for every countably infinite subset \(A\) of \(X\), there exists a countable subset \(B\) of \(X\) such that \(A\subseteq B\) and there exists a condensation of \(X\setminus B\) onto a compact Hausdorff space. A compact weakly \(a\)-subcompact space is called a weak \(a\)-space. A Hausdorff space is dyadic if it is a continuous image of a Cantor cube. The authors generalize their result from [\textit{V. I. Belugin} et al., Math. Notes 109, No. 6, 849--858 (2021; Zbl 1473.54010)] by proving that, for every compact Hausdorff space \(X\) and every dense-in-itself dyadic Hausdorff space \(Y\), the Cartesian product \(X\times Y\) is a strict \(a\)-space. Furthermore, among other facts, the authors show what follows. If a Hausdorff dyadic space \(Y\) has an isolated point, and \(X\) is a compact Hausdorff space which is not a strict \(a\)-space, then the product \(X\times Y\) is not a strict \(a\)-space. If \(X\) is a compact Hausdorff space such that \(X\times (\omega_1+1)\) is an \(a\)-space, then \(X\) is a strict \(a\)-space. There exists a pair \(X, Y\) of \(a\)-spaces whose product \(X\times Y\) is not an \(a\)-space. Every compact metrizable space is a strict \(a\)-space. If there exists a condensation of a Hausdorff space \(X\) onto a compact metrizable space, then \(X\) is strictly \(a\)-subcompact. Every second-countable rimcompact (called also peripherally compact) metrizable space which is an absolute \(G_{\delta}\) is strictly \(a\)-subcompact. Every dyadic Hausdorff space is a strict \(a\)-space. For every dyadic Hausdorff space \(X\) and every countable subset \(C\) of \(X\), there is a condensation of \(X\setminus C\) onto a dyadic Hausdorff space. If there is a condensation of a Hausdorff space \(X\) onto a dyadic Hausdorff space, then \(X\) is strictly \(a\)-subcompact. If \(X\) is a weakly \(a\)-subcompact space and \(Y\) is a compact Hausdorff countable space, then \(X\times Y\) is weakly \(a\)-subcompact. There exists a weak \(a\)-space which is not an \(a\)-space. The authors have solved several older open problems and posed new open problems concerning the above-mentioned notions.