An Extension of Galvin's Theorem Related to the Banach-Mazur Game

Let $X$ be a topological space. Galvin’s theorem on the Banach-Mazur game states that if $X$ has a $\pi$-base $\mathcal{B}$ satisfying the following condition: $\mathit{\text{For}}\mathit{\text{any}}U\in \mathcal{B}\mathit{\text{,}}\mathit{\text{there}}\mathit{\text{exists}}\mathit{\text{a}}\mathit{\text{disjoint}}\mathit{\text{family}}{\mathcal{A}}_U\mathit{\text{of}}\mathit{\text{nonempty}}\mathit{\text{open}}\mathit{\text{subsets}}\mathit{\text{of}}U\mathit{\text{such}}\mathit{\text{that}}$ $$\mathrm{\mid }{\mathcal{A}}_U\mathrm{\mid }\ge \mathrm{\mid }\{B\in \mathcal{B}:U\subseteq B\}\mathrm{\mid },$$ 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 ${\mathcal{A}}_U$, we showed that, the theorem is also true.