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An Extension of Galvin's Theorem Related to the Banach-Mazur Game
Abstract
Let X be a topological space. Galvin's theorem on the Banach-Mazur game states that if X has a π-base ℬ satisfying the following condition: \textit{For any U ∈ ℬ, there exists a disjoint family 𝒜U of nonempty open subsets of U such that} |𝒜U| ≥ |{ B ∈ ℬ : U ⊆ B }|, then, in the Banach-Mazur game on X, the second player, denoted NONEMPTY, has a 2-tactic if NONEMPTY has a winning strategy in this game. In this paper, we extend the disjointness condition of this family by replacing it with the condition of being open finite family. With this extension of the family 𝒜U, we showed that, the theorem is also true.
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BibTeX
@misc{soyarslan2026extension,
title = {{An Extension of Galvin's Theorem Related to the Banach-Mazur Game}},
author = {Servet Soyarslan and Süleyman Önal},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26643},
eprint = {2610.26643},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26643},
}
RIS
TY - UNPB
TI - An Extension of Galvin's Theorem Related to the Banach-Mazur Game
AU - Servet Soyarslan
AU - Süleyman Önal
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26643
UR - https://arxiv.org/abs/2610.26643
AB - Let X be a topological space. Galvin's theorem on the Banach-Mazur game states that if X has a π-base ℬ satisfying the following condition: \textit{For any U ∈ ℬ, there exists a disjoint family 𝒜_U of nonempty open subsets of U such that} |𝒜_U| ≥ |{ B ∈ ℬ : U ⊆ B }|, then, in the Banach-Mazur game on X, the second player, denoted NONEMPTY, has a 2-tactic if NONEMPTY has a winning strategy in this game. In this paper, we extend the disjointness condition of this family by replacing it with the condition of being open finite family. With this extension of the family 𝒜_U, we showed that, the theorem is also true.
ER -