{"id":"palomar/PALOMAR-2026-10-21-000013","url":"https://github.com/Arthur742Ramos/Math-Formalization/tree/0377aacab289fee8153cdec87fd8a1be06b9ec2c/topology/banach-mazur-category","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-21-000013-v1.json","repo":"https://github.com/Arthur742Ramos/Math-Formalization","tags":["math.GN","msc:54E52","msc:91A44","msc:03E15","set-theory","topology","game-theory","baire-category"],"score":0,"title":"Banach–Mazur determinacy for Baire-property targets","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":"For any complete metric space and any target set with the property of Baire, the Banach–Mazur category game is determined: the first player has a winning full-history strategy exactly when the target is nonmeagre, and the second exactly when it is meagre. A win for the first player means one target point lies in every move. The proof constructs meagre-target avoidance without metric assumptions and shrinking closed-trap target forcing in complete metric spaces; it proves actual joint-play existence and excludes simultaneous winners. Empty spaces and equal nested moves are explicit. This is…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-21","reactions":{},"formalized":true,"formalizes":[{"title":"Banach-Mazur Games with Simple Winning Strategies, CSL2012, pp.305–319","identifier":"https://doi.org/10.4230/LIPIcs.CSL.2012.305","relationship":"formalizes"}]}