There are no hereditary productive \(\gamma \)-spaces
Hereditarily Lindelöf topological spaces in which every open set is a countable union of clopen sets are the subject of investigation. The main result says that every uncountable productive \(\gamma\)-space contains a co-countable subspace which is not Hurewicz, and thus not a \(\gamma\)-space. Several results interesting by their own are proved on the way to the main result. Every co-countable subspace of \(X\) is Hurewicz if and only if \(X\) is a Hurewicz \(\lambda\)-space. Unions of countably many productive \(\gamma\)-spaces are productive \(\gamma\)-spaces. A question of Miller is answered by showing that unions of increasing sequences of \(\gamma\)-spaces are \(\gamma\)-spaces. The Hurewicz \(\lambda\)-spaces \(X\) are characterized in terms of some continuous functions \(f:X\to (\omega\cup\{\infty\})^{\omega}\). Results concerning preservation with respect to unions are based on a theorem which gives that the suprema of (increasing) countable sequences of (productive) Fréchet filters are (productive) Fréchet filters.
scientific article; zbMATH DE number 5346280