Filters, consonance and hereditary Baireness
For a topological space \(X\), let \(F(X)\) be the family of all closed subsets of \(X\). For \(F(X)\) the co-compact topology \(\tau_{\text{co}}\) is the topology generated by the family of all subsets of the form \(\{F\in F(X):F\cap K=\emptyset\}\), where \(K\) is an arbitrary compact subset of \(X\), and the upper Kuratowski topology \(\tau_{uk}\) is the topology associated with the upper Kuratowski-Painlevé convergence, where a net \((F_\gamma)_{\gamma \in\Gamma}\subset F(X)\) upper Kuratowski-Painlevé converges to \(F\in F(X)\) means that \(\bigcap\{\overline {\bigcup\{F_\gamma: \gamma\geq \alpha}\}: \alpha\in \Gamma\}\subset F\). The space \(X\) is said to be consonant if the equality \(\tau_{uk}= \tau_{\text{co}}\) holds. For a filter \({\mathcal F}\) on the set of natural numbers \(\omega\), let \(X_{\mathcal F}=\omega \cup\{\infty\}\) be the space for which all points in \(\omega\) are isolated and the neighborhood system of \(\infty\) is \(\{A\cup\{\infty\}: A\in{\mathcal F}\}\). The author gives a combinatorial characterization of the class \(\Phi\) of all filters \({\mathcal F}\) such that the space \(X_{\mathcal F}\) is consonant and all its compact subsets are finite. It is also shown that a filter \({\mathcal F}\) belongs to \(\Phi\) if and only if the space \(C_p(X_{\mathcal F})\) of real-valued continuous functions on \(X_{\mathcal F}\) with the pointwise-convergence topology is hereditarily Baire. Let \(\beta\omega\) denote the Čech-Stone compactification of the discrete space \(\omega\) and \(\omega^*= \beta\omega \smallsetminus\omega\). For a filter \({\mathcal F}\) on \(\omega\), let \(A({\mathcal F})\) be the subset of \(\omega^*\) of all ultrafilters on \(\omega\) finer than \({\mathcal F}\). \({\mathcal F}\) is said to be a \(P\)-filter if \(A({\mathcal F})\) is a \(P\)-set of \(\omega^*\); equivalently, for every sequence \((A_n)_{n\in \omega}\subset {\mathcal F}\) there is \(A\in{\mathcal F}\) such that \(A\subseteq^*A\) (that is, \(A\smallsetminus A_n\) is finite) for each \(n\in \omega\). The main results are stated below: Theorem. Let \({\mathcal F}\) be a filter on \(\omega\) endowed with the subspace topology of the Cantor set \(\{0, 1\}^\omega\). Then the following are equivalent: (1) \(X_{\mathcal F}\) is consonant and all its compact subsets are finite. (2) \({\mathcal F}\) is a \(P\)-filter and \(C_p (X_{\mathcal F})\) is a Baire space. (3) \({\mathcal F}\) is a \(P\)-filter and \({\mathcal F}\) is a Baire space. (4) \(C_p(X_{\mathcal F})\) is hereditarily Baire. (5) \({\mathcal F}\) is hereditarily Baire. (6) \({\mathcal F}\in\Phi\).
scientific article; zbMATH DE number 1462691