Cut-and-choose games in topological spaces
Lucas Chiozini, Tamás Csernák, Lajos Soukup
TL;DR
This paper studies transfinite cut-and-choose games on $T_0$ spaces by introducing the point-separating number $ps(X)$ and the set membership number $sm(X)$ as ordinal-valued invariants capturing the minimal game length needed to determine a hidden point or subset. It establishes fundamental inequalities linking these invariants with $\psi w_0(X)$ and $|X|$, provides sharpness examples, and computes the invariants for classical spaces such as Cantor cubes, powers of the Alexandroff double arrow space, and stationary subsets. The authors analyze how $ps$ and $sm$ behave under topological sums and products, highlighting a striking contrast between the two invariants. They show that for metric (and developable) spaces $ps(X)=\log|X|$, while $sm(X)$ can be arbitrarily large; they also develop a stepping-up technique to realize a broad range of ordinals as values of $ps$, and discuss several open problems, including determinacy of these games and the precise values for Suslin lines and $\omega^*$. Overall, the work deepens understanding of ordinal-valued topological invariants arising from informational guessing games and reveals intricate interactions with classical topological invariants.
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 $ψw_0$. We establish fundamental inequalities relating $ps(X)$, ${sm}(X)$, $ψw_0(X)$, and $|X|$, including the sharp bounds $|X|\le 2^{ps(X)}$ and $ψw_0(X)\le 2^{<ps(X)}$. 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.