Abstract
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.
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Galvin, F., Scheepers, M. Baire spaces and infinite games. Arch. Math. Logic 55, 85–104 (2016). https://doi.org/10.1007/s00153-015-0461-8
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DOI: https://doi.org/10.1007/s00153-015-0461-8