Dual Selection Games

Abstract

Often, a given selection game studied in the literature has a known dual game. In dual games, a winning strategy for a player in either game may be used to create a winning strategy for the opponent in the dual. For example, the Rothberger selection game involving open covers is dual to the point-open game. This extends to a general theorem: if $\{\mathrm{ran}f:f\in \mathbf{C}(\mathcal{R})\}$ is coinitial in $\mathcal{A}$ with respect to $\subseteq$, where $\mathbf{C}(\mathcal{R})=\{f\in (\bigcup \mathcal{R}{)}^{\mathcal{R}}:R\in \mathcal{R}\Rightarrow f(R)\in R\}$ collects the choice functions on the set $\mathcal{R}$, then $G_1(\mathcal{A},\mathcal{B})$ and $G_1(\mathcal{R},\neg \mathcal{B})$ are dual selection games.