Remarks on the space \(\aleph_{1}\) in \(\mathbf {ZF}\)
A space \(X\) is said to be effectively normal if there exists a function \(F\) associating to every pair \(A, B\) of disjoint closed sets in \(X\) a pair \(C, D\) of disjoint open sets including \(A, B\) respectively. The authors prove in this paper that \(\aleph_{1}\) with the order topology is effectively normal. It is also proved that for every countable ordinal \(\alpha\), the ordered space \(\alpha\) is metrizable and so every closed subset of \(\alpha\) is a zero set and therefore the Čech-Stone extension of \(\alpha\) coincides with its Wallman extension. The Wallman extension for a space \(X\) is defined as follows: Let \(C\) be a \(T_1\) base for \(X\). \(\mathcal W(X,\mathcal C)\) denotes the set of all maximal \(C\)-filters \(\mathcal F\). The topology on \(\mathcal W(X,\mathcal C)\) having as a base for the closed sets the family \(\{A^{*}|A\in \mathcal C\}\) where \(A^{*}=\{\mathcal F \in \mathcal W(X,C):A\in \mathcal F\}\) is called the Wallman topology on \(\mathcal W(X,\mathcal C)\). The set \(\mathcal W(X,\mathcal C)\) with the Wallman topology is called the Wallman extension of \(X\) corresponding to \(\mathcal C\). If \(\mathcal C\) is the set \(\mathcal Z\) of all zero sets of \(X\), the Wallman extension is the Čech-Stone extension \(\beta X\) of \(X\). The authors prove that in the Feferman-Levy model where \(\aleph_{1}\) is singular, the ordinal space \(\aleph_{1}\) is base-Lindelöf but not Lindelöf. It is proved that \(\beta \aleph_{1}\) is compact iff its Wallman extension \(\mathcal W\aleph_1\) is compact. It is a nice result proved by the authors in this paper that the set \(L\) of all limit ordinals of \(\aleph_{1}\) is closed but not a zero set.
scientific article; zbMATH DE number 5835750