Baire Spaces and Infinite Games
It is well known that if the nonempty player of the Banach–Mazur game has a winning strategy on a space, then that space is Baire in all powers even in the box product topology. The converse of this implication may also be true: We know of no consistency result to the contrary. In this paper we establish the consistency of the converse relative to the consistency of the existence of a proper class of measurable cardinals.
Galvin, Fred and Scheepers, Marion. (2016). "Baire Spaces and Infinite Games". Archive for Mathematical Logic, 55(1-2), 85-104. https://doi.org/10.1007/s00153-015-0461-8