A note on the Arens' space and sequential fan
A cover \({\mathcal P}\) of a space is a \(k\)-network if, for any compact subset \(K\) and any open subset \(U\) with \(K\subset U\), there exists a finite \({\mathcal P}^*\subset{\mathcal C}\) such that \(K\subset \bigcup{\mathcal P}^*\subset U\). Lashnev spaces, quotient \(s\)-images of metric spaces, or CW-complexes are \(k\)-spaces with a point-countable \(k\)-network. This paper obtains the following results and others by means of sequential coreflections \(\sigma(X)\). The results below improve the reviewer's results, where spaces have \(G_\delta\)-points. Here, let us assume that spaces are regular and \(T_1\). Theorem: Let \(X\) be a \(k\)-space with a point-countable \(k\)-network. Then, \(X\) is a Fréchet space iff \(X\) contains no closed copy of the Arens' space \(S_2\). Corollary: Let \(X\) be a \(k\)-space with a point-countable \(k\)-network. Then, (a) \(X\) has a point-countable base iff \(X\) contains no closed copy of \(S_2\), and no sequential fan \(S_\omega\). (b) \(X^\omega\) is a \(k\)-space iff \(X\) has a point-countable base.
scientific article; zbMATH DE number 1103050
scientific article