{"id":"palomar/PALOMAR-2026-10-20-000008","url":"https://github.com/Arthur742Ramos/Math-Formalization/tree/59d4a50df466480e88ab71bc1f3527bfe0bf1811/topology/banach-mazur-baire","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-20-000008-v1.json","repo":"https://github.com/Arthur742Ramos/Math-Formalization","tags":["math.GN","msc:54E52","msc:91A44","general-topology","baire-spaces","banach-mazur-game","game-theory"],"score":0,"title":"Oxtoby's Banach–Mazur characterization of Baire spaces","trust":"high","domain":"math.GN","source":"palomar","authors":["Arthur Freitas Ramos","David Barros Hulak","Ruy Jose Guerra Barretto de Queiroz"],"reviews":0,"subject":"math.GN","summary":"An arbitrary topological space is Baire if and only if the first player has no winning full-history strategy in the nested nonempty-open Banach–Mazur game. The proof constructs a compatible play with one common point from disjoint dense response levels, and constructs a winning strategy from failure of Baireness. It separately verifies compatible-play existence, equality of the two players’ intersections and the empty-space boundary. No separation, countability or metric hypothesis is imposed. This is classical mathematics, not a novelty or determinacy claim.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-20","reactions":{},"formalized":true,"formalizes":[{"title":"More on products of Baire spaces, arXiv:1702.06197v3","identifier":"https://arxiv.org/pdf/1702.06197v3","relationship":"formalizes"},{"title":"Some variations of the Banach-Mazur game, arXiv:2104.02269v2","identifier":"https://arxiv.org/pdf/2104.02269v2","relationship":"formalizes"}]}