Topological spaces that are $\alpha$-favorable for a player with perfect information

The class of spaces mentioned in the title is closely related to the class of $\alpha$-favorable spaces introduced by G. Choquet [3]. For convenience, call the spaces mentioned in the title weakly $\alpha$-favorable. The following statements are true: (1) every dense ${G_\delta }$ subset of a quasi-regular, weakly $\alpha$-favorable space is weakly $\alpha$-favorable; (2) the product of any family of weakly $\alpha$-favorable spaces is weakly $\alpha$-favorable; (3) any continuous, open image of a weakly $\alpha$-favorable space is weakly $\alpha$-favorable; (4) a quasi-regular space with a $\sigma$-disjoint pseudo-base is weakly $\alpha$-favorable if and only if it is pseudo-complete in the sense of J. C. Oxtoby; and (5) the product of a weakly $\alpha$-favorable space and a Baire space is a Baire space.