On hereditarily normal compact spaces, in which regular closed subsets are \(G_ \delta\)-sets (Q1329767)

A topological space \(X\) is \(k\)-perfect if every regular closed subset of \(X\) is a \(G_ \delta\)-set. Some properties of hereditarily normal \(k\)- perfect spaces are considered. Let \({\mathcal K}\) denote the class of all hereditarily normal \(k\)-perfect compact spaces. Every \(X \in {\mathcal K}\) has countable tightness and, under \(2^ \omega = \omega_ 2\), countable spread. The assertion ``every \(X\) in \({\mathcal K}\) is perfectly normal'' is independent of and consistent with ZFC: under \(\diamondsuit\) there exists a space \(X\) in \({\mathcal K}\) which is not perfectly normal (Theorem 2.3), while under \(MA + 2^ \omega = \omega_ 2\) every \(X \in {\mathcal K}\) is perfectly normal (Theorem 3.6).