Fréchet--Urysohn for finite sets
This paper provides a variety of results and examples related to topological spaces which are Fréchet-Urysohn. Reznichenko and Sipacheva defined a space \(X\) to be \textit{Fréchet-Urysohn for finite sets} abbreviated \(FU_{\text{fin}}\) if for each \(x\) in \(X\) and collection \(P\) of finite subsets of \(X\) with every neighborhood of \(x\) containing a member of \(P\), a subfamily of \(P\) converges to \(x\). The authors consider spaces called \textit{boundedly-}\(FU_{\text{fin}}\) which limits the choices of \(P\) . It is shown that although each \(FU_{\text{fin}} \) space is boundedly-\(FU_{\text{fin}}\) , there are spaces which are boundedly-\(FU_{\text{fin}}\) but not \(FU_{\text{fin}}\). The study includes various examples of \(FU_{\text{fin}}\) spaces related to \(2^{\omega}\) and \(\omega^{\omega}\) and relations to \(\alpha_i\)-spaces introduced by Arhangel'skii. A natural association between \(FU_{\text{fin}}\) spaces and Fréchet-Urysohn groups is explored. Further, \(FU_{\text{fin}}\) spaces are characterized in terms of game theory and consequences for cartesian products are investigated.
scientific article; zbMATH DE number 2174918
scientific article