Cut-and-choose games in topological spaces

Abstract

We study transfinite cut-and-choose games on $T_0$ spaces, introducing the {\em point-separating number} $ps(X)$ and the {\em set membership number} ${sm}(X)$ as the ordinal-valued invariants measuring the minimal length of a game in which a Seeker can determine a hidden point or subset. A central motivating question is which countable ordinals can occur as the value of $ps(X)$, in particular whether any countable ordinal can arise. These invariants generalize Scott's $T_0$-pseudoweight $\psi w_0$. We establish fundamental inequalities relating $ps(X)$, ${sm}(X)$, $\psi w_0(X)$, and $|X|$, including the sharp bounds $|X|\le 2^{ps(X)}$ and $\psi w_0(X)\le 2^{<ps(X)}$. We compute these invariants for familiar spaces such as Cantor cubes, powers of the Alexandroff double arrow space, and certain stationary subsets of cardinals. We further investigate their behavior under topological sums and products, revealing the striking contrast between $ps$ and ${sm}$. For metric spaces, we determine that $ps(X)=\log|X|$. However, we do not know such computation for ${sm}(X)$; we can only assert that ${sm}(X)$ may be arbitrarily large. Finally, we highlight another open problem: whether these games are always determined.