Fréchet–Urysohn for finite sets, II

We continue our study [G. Gruenhage, P.J. Szeptycki, Fréchet Urysohn for finite sets, Topology Appl. 151 (2005) 238–259] of several variants of the property of the title. We answer a question from that paper by showing that a space defined in a natural way from a certain Hausdorff gap is a Fréchet α 2 space which is not Fréchet–Urysohn for 2-point sets (FU2), and answer a question of Hrušák by showing that under MA ω 1 , no such “gap space” is FU2. We also introduce versions of the properties which are defined in terms of “selection principles”, give examples when possible showing that the properties are distinct, and discuss relationships of these properties to convergence in product spaces, to the α i -spaces of A.V. Arhangelʹskii, and to topological games.