Fréchet-Urysohn for finite sets

E. Reznichenko and O. Sipacheva called a space X “Fréchet-Urysohn for finite sets” if the following holds for each point x∈X: whenever P is a collection of finite subsets of X such that every neighborhood of x contains a member of P, then P contains a subfamily that converges to x. We continue their study of this property. We also look at analogous notions obtained by restricting to collections P of bounded size, we discuss connections with topological groups, the αi-properties of A.V. Arhangel'skii, and with a certain topological game.