@article{Galvin2016BaireSA,
title={Baire spaces and infinite games},
author={F. Galvin and M. Scheepers},
journal={Archive for Mathematical Logic},
year={2016},
volume={55},
pages={85-104}
}

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.