Distinguishing perfect set properties in separable metrizable spaces (Q2805029)

In this article, the author investigates the perfect set property in separable metrizable spaces which are not neccesarily Polish.NEWLINENEWLINELet \(X\) be a separable metrizable space and \({\Gamma}\) a pointclass. We say \(X\) has the \textsl{perfect set property} for \(\Gamma\) sets, denoted by PSP(\({\Gamma}\)), if every \(\Gamma\) subset of \(X\) either is countable or contains a copy of \(2^\omega\).NEWLINENEWLINEThe author proves the following results: PSP(coanalytic) \(\Rightarrow\) PSP(analytic) \(\Leftrightarrow\) PSP(\(G_\delta\)) \(\Rightarrow\) PSP(\(F_\sigma\)) \(\Leftrightarrow\) PSP(closed) \(\Rightarrow \) PSP(open) \(\Rightarrow\) PSP(clopen). Counterexamples are also presented to show that: PSP(conanalytic) \(\not\Leftarrow\) PSP(analytic), PSP(closed) \(\not\Leftarrow\) PSP (open), and PSP(open) \(\not\Leftarrow\) PSP(clopen).NEWLINENEWLINEThe highlight of this article is to show that the statement PSP(closed)\(\Rightarrow\) PSP(analytic) holds for all separable metrizable spaces iff \({\mathfrak b}>\omega_1\), where \(\mathfrak b\) is the least size of an unbounded family in \((\omega^\omega,<^*)\). Thus the statement PSP(closed) \(\Rightarrow\) PSP(analytic) is independent of ZFC. In proving this, a technical notion is introduced as follows:NEWLINENEWLINEWe say a subset \(W\subseteq 2^\omega\) has the \textsl{Grinzing property} if it is uncountable and, for any uncountable \(Y\subseteq W\), there exists an uncountable collection of uncountable subsets of \(Y\) with pairwise disjoint closures in \(2^\omega\). The following results are presented: {\parindent=0.7cm\begin{itemize}\item[(1)] There exists a subset of \(2^\omega\) with the Grinzing property. \item[(2)] Assume \(\mathrm{MA}+\neg \mathrm{CH}\). Then \(2^\omega\) has the Grinzing property. \item[(3)] Assume CH. Then \(2^\omega\) does not have the Grinzing property. NEWLINENEWLINE\end{itemize}}NEWLINENEWLINEGeneralizations of the Grinzing property and the perfect set property are also considered in the rest of the article.