Fréchet-Urysohn for finite sets, II (Q2381641)

This paper continues the study begun in [Topology Appl. 151, 238--259 (2005; Zbl 1085.54016)]. It is established that there exists a space which is Fréchet \(\alpha _2\) but not Fréchet-Urysohn for 2-point sets and hence not \(FU_{\text{fin}}\) answering a question posed in the paper cited above. The example relates to a Hausdorff gap space. The paper also extends relationships of \(FU_{\text{fin}}\) with ``selection principles'' , topological games, and the \(\alpha_i\)-spaces introduced by Arhangel'skii.