Subsets of \({}^ \omega\omega\) and the Fréchet-Urysohn and \(\alpha_ i\)-properties
A space is Fréchet-Urysohn if \(x\) in the closure of \(A\) implies \(A\) has a subsequence converging to \(x\). Arkhangelskij defined subclasses by adding the restriction that the spaces have one of the so-called \(\alpha_ i\)-properties \((0\leq i\leq 4)\) in order to get better behaviour (in products for example). A countable Fréchet-Urysohn \(\alpha_ 2\)-space which is not first countable \((\alpha_ 0)\) is constructed. Under a variety of additional set-theoretic hypotheses, the classes \(\alpha_ 2\), \(\alpha_ 1\) and \(\alpha_ 0\) are distinguished. It is now known that it is consistent that \(\alpha_ 2\) implies \(\alpha_ 1\) and \(\alpha_ 1\) implies \(\alpha_ 0\), but not both simultaneously as the above mentioned example demonstrates. An example of a countable \(\alpha_ 2\)-topological group that is not first countable is given. It is shown to be Fréchet-Urysohn if \({\mathfrak p}={\mathfrak b}\). It is unknown if it is consistent that each countable Fréchet-Urysohn topological group is first countable (hence metrizable).
scientific article; zbMATH DE number 166414
scientific article