A variation of the proximal infinite game (Q1637108)

All spaces are assumed to be Hausdorff. The author considers a variation of the proximal infinite game, which leads to the introduction of a class of spaces more general than the proximal spaces. If the first player has a winning strategy in this variation, then the space is pseudonormal. If the second player does not have a winning strategy, then the space is an Arhangel'skii \(\alpha_2\) space. The new version of the proximal game is used to prove that a \(\Sigma\)-product of \(\omega\)-bounded topological manifolds is pseudonormal. A space is said to be pseudonormal if each pair of disjoint closed sets can be separated by disjoint open sets provided that at least one of the closed sets is countable.