On classes of subcompact spaces (Q2037734)

This is one more study of the classical problem of Alexandroff on the existence of a continuous bijection onto a compact space: when can a Hausdorff space \(X\) be mapped onto a compact space by a one-to-one continuous mapping? Such spaces \(X\) are called subcompact. The authors define \(a_{\tau}\)-spaces and strictly \(a_{\tau}\)-spaces, where \(\tau\) is a cardinal, as a higher cardinal generalization of \(a\)-spaces and strictly \(a\)-spaces. A compact space \(X\) is an \(a_{\tau}\)-space (resp., strictly \(a_{\tau}\)-space) if for any \(C\subset X\) of cardinality \(\le \tau\) there is a continuous bijection from \(X\setminus C\) onto a compact space \(Y\) (resp., there is a continuous bijection from \(X\setminus C\) onto a compact space \(Y\) that can be continuously extended to \(X\)). Several results on the existence and properties of these spaces are obtained. For example: An infinite product of non-one-point compact spaces is a strictly \(a_{\tau}\)-space. Six open problems are posed.