We show that a recent existence result for the Nash equilibria of generalized games with strategy sets in -spaces is false. We prove that it becomes true if we assume, in addition, that the feasible set of the game (the set of all feasible multistrategies) is closed.
We present a negative answer to problem 3.7(b) posed on page 193 of [2], where, in fact, A. Rosłanowski asked: Does every set of Lebesgue measure zero belong to some Mycielski ideal?
Answering a question of Telgársky in the negative, it is shown that there is a space which is β-favorable in the strong Choquet game, but all of its nonempty -subspaces are of the second category in themselves.
We modify a game due to Berner and Juhász to get what we call “the open-open game (of length ω)”: a round consists of player I choosing a nonempty open subset of a space X and II choosing a nonempty open subset of I’s choice; I wins if the union of II’s open sets is dense in X, otherwise II wins. This game is of interest for ccc spaces. It can be translated into a game on partial orders (trees and Boolean algebras, for example). We present basic results and various conditions under which I or II...
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.