Let E be an infinite dimensional separable space and for e ∈ E and X a nonempty compact convex subset of E, let qX(e) be the metric antiprojection of e on X. Let n ≥ 2 be an arbitrary integer. It is shown that for a typical (in the sence of the Baire category) compact convex set X ⊂ E the metric antiprojection qX(e) has cardinality at least n for every e in a dense subset of E.
In the main result, partially answering a question of Telgársky, the following is proven: if X is a first countable R₀-space, then player β (i.e. the EMPTY player) has a winning strategy in the strong Choquet game on X if and only if X contains a nonempty -subspace which is of the first category in itself.
We continue the study of almost--resolvable spaces beginning in A. Tamariz-Mascar’ua, H. Villegas-Rodr’ıguez, Spaces of continuous functions, box products and almost--resoluble spaces, Comment. Math. Univ. Carolin. 43 (2002), no. 4, 687–705. We prove in ZFC: (1) every crowded space with countable tightness and every space with -weight is hereditarily almost--resolvable, (2) every crowded paracompact space which is the closed preimage of a crowded Fréchet space in such a way that the...