The cardinal characteristic for relative \(\gamma \)-sets (Q1009733)

This is a paper about cardinals, at most continuum (\(\mathfrak c = 2^{\aleph_0}\)), that arise in connection with covering properties of separable metric spaces. An \(\omega\)-cover of a space is a collection of its open subsets, such that each finite set of points is contained in some set in the collection. By a \(\gamma\)-cover of a space is meant a sequence \(\langle U_n : n <\omega\rangle\) such that each point is contained in cofinitely many \(U_n\). Clearly the sets constituting a \(\gamma\)-cover also form an \(\omega\)-cover; a subset \(X\) of a separable metric space \(Y\) is a relative \(\gamma\)-set in \(Y\) if each \(\omega\)-cover of \(Y\) refines to a collection that serves as a \(\gamma\)-cover of \(X\). \(X\) is simply a \(\gamma\)-set if \(X\) is a relative \(\gamma\)-set in itself. Relative \(\gamma\)-sets were introduced in [\textit{Lj. D. R. Kočinac, C. Guido} and \textit{L. Babinkostova}, East-West J. Math. 2, No.~2, 195--199 (2000; Zbl 0966.54009)] as a generalization of the \(\gamma\)-sets initially studied by \textit{J. Gerlits} and \textit{Zs. Nagy} [Topology Appl. 14, 151--161 (1982; Zbl 0503.54020)]. In the earlier paper it was shown that: (1) non-\(\gamma\)-sets exist; and (2) the smallest cardinality of a non-\(\gamma\)-set equals the pseudo-intersection number \(\mathfrak p\), the smallest cardinality of a collection \(\mathcal F\) of subsets of \(\omega\) such that each intersection of a finite subfamily of \(\mathcal F\) is infinite, but no infinite \(X\subseteq\omega\) has the property that \(X \backslash Y\) is finite for all \(Y\in\mathcal F\); i.e., \(\mathcal F\) has no pseudo-intersection. (Nonprincipal ultrafilters, for example, have no pseudo-intersection.) For \(\mathcal F\) a free filter -- i.e., one containing all cofinite sets -- on \(\omega\), define \(\mathfrak p_{\mathcal F}\) to be the smallest cardinality of a subset of \(\mathcal F\) having no pseudo-intersection. Also, for a separable metric \(Y\), define \(\mathfrak p(Y)\) to be the smallest cardinality of a non-\(\gamma\)-set relative to \(Y\). As usual, \(\omega^\omega\) and \(2^\omega\) represent the irrational reals and the Cantor set respectively, and it is easy to show that \(\mathfrak p\leq \mathfrak p(\omega^\omega)\leq \mathfrak p(2^\omega)\leq\mathfrak c\). Here are some of the main results of the present paper. {\parindent6mm \begin{itemize}\item[(1)] \(\mathfrak p(\omega^\omega)\) is the minimum of the \(\mathfrak p_{\mathcal F}\) such that \(\mathcal F\) is a \(\Sigma_1^1\) free filter on \(\omega\). \item[(2)] \(\mathfrak p(2^\omega)\) is the minimum of the \(\mathfrak p_{\mathcal F}\) such that \(\mathcal F\) is a \(\Sigma_2^0\) free filter on \(\omega\). \item[(3)] If \(X\) is an uncountable \(\Sigma_1^1\) set in a Polish space, then \(\mathfrak p(X)\) is either \(\mathfrak p(2^\omega)\) or \(\mathfrak p(\omega^\omega)\) (depending on whether or not \(X\) is \(\sigma\)-compact). \item[(4)] The statements ``\(\mathfrak p = \mathfrak p(\omega^\omega) < \mathfrak p(2^\omega)\)'' and ``\(\mathfrak p < \mathfrak p(\omega^\omega) = \mathfrak p(2^\omega)\)'' are both consistent with ZFC. \end{itemize}}