Countable Fréchet \(\alpha_ 1\)-spaces may be first countable
An \(\alpha_ 1\)-space is one in which the following condition holds: If \(\{S_ n\): \(n\in\omega\}\) is a countable collection of sequences converging to a point \(x\) then there is one sequence \(S\) that converges to \(x\) and is such that \(S_ n\subseteq^*S\) for all \(n\). It should be clear that first-countable spaces are \(\alpha_ 1\); the converse is not true as witnessed by a consistent examples of a Fréchet-Urysohn \(\alpha_ 1\)-space that is not first-countable [the first author, Proc. Am. Math. Soc. 108, No. 1, 241-247 (1990; Zbl 0675.54029)]. The authors construct a model of set theory in which all countable Fréchet-Urysohn \(\alpha_ 1\)-spaces are first-countable, thus eliminating the most obvious candidates for counterexamples, namely spaces of the form \(\omega\cup \{x\}\). The result may also be interpreted in terms of the space \(\omega^*\). For consider the trace \({\mathcal F}\) of the neighbourhood filter of \(x\) on \(\omega\) and its associated closed subset \(F=\bigcap \{A^*\): \(A\in {\mathcal F}\}\) of \(\omega^*\). It is not too hard to check that \(\omega\cup \{x\}\) is first-countable iff \(F\) is a \(G_ \delta\)-set; it is Fréchet-Urysohn iff \(F\) is regularly closed and that it is \(\alpha_ 1\) iff for every countable family \({\mathcal O}\) of clopen subsets of \(\omega^*\) such that \(\bigcup {\mathcal O} \subseteq F\) there is one clopen set \(C\) such that \(\bigcup {\mathcal O} \subseteq C\subseteq F\). We see that in the authors' model the \(G_ \delta\)-subsets of \(\omega^*\) have an -- as it were -- internal characterization: A set is a \(G_ \delta\)-set iff it is regularly closed and the family if its clopen subsets if countably directed. With every filter \({\mathcal F}\) on \(\omega\) the authors associate a partial order \(\mathbb{M}({\mathcal F})\) which shoots an infinite subset of \(\omega\) below the filter \({\mathcal F}\). If the \({\mathcal F}\) is Fréchet-Urysohn and \(\alpha_ 1\) then \(\mathbb{M}({\mathcal F})\) is proper and satisfies a condition that generally guarantees that certain extremal collections of subsets of \(\omega\) remain extremal after forcing with \(\mathbb{M}({\mathcal F})\) (the authors call such partial orders hyper-proper). The model is constructed in an \(\omega_ 2\)-stage countable support iteration and a \(\diamondsuit\)-sequence on \(D= \{\alpha\in\omega_ 3\): cf. \(\alpha=\omega_ 1\}\) is used to capture (names of) potential non first-countable Fréchet-Urysohn \(\alpha_ 1\)-filters. If, in the final model, \({\mathcal F}\) were a counterexample then it would have been predicted cofinally often on \(D\); after the first \(\omega\) many times the infinite sets forced by the \(\mathbb{M}({\mathcal F})\)'s have destroyed \(\alpha_ 1\)-ness of the first prediction of \({\mathcal F}\) and hyper-properness ensures that it is lost forever. Sadly the exposition of this nice piece of mathematics is marred by an inordinate number of misprints and inaccuracies.
scientific article; zbMATH DE number 484791