Leírás
The speaker of the next Set Theory Seminar will be Lajos Soukup, presenting joint work with Tamás Csernák and Lucas Chiozini de Souza.
Date: October 9, 2025
Time: 10:30 AM – 12:00 PM (Central European Time)
Location: HUN-REN Rényi Institute, Nagyterem, Budapest, Hungary
Title: Cut-and-choose games in topological spaces
Abstract:
We study transfinite cut-and-choose games on $T_0$ spaces, introducing the point-separating number $\ps(X)$ and the set membership number ${\sm}(X)$ as ordinal-valued invariants that measure 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 $\psn(X)$.
We establish fundamental inequalities relating $\ps(X)$, ${\sm}(X)$, $\psn(X)$, and $|X|$, including the sharp bounds $|X| \le 2^{\ps(X)}$ and $\psn(X) \le 2^{<\ps(X)}$. We compute these invariants for familiar spaces such as Cantor cubes, powers of the Alexandroff double arrow, and certain stationary subsets of cardinals.
We further investigate their behavior under topological sums and products, revealing a striking contrast between $\ps$ and ${\sm}$.
For metric spaces, we determine that $\ps(X) = \log|X|$. However, the corresponding value of ${\sm}(X)$ remains unknown; we can only assert that ${\sm}(X)$ may be arbitrarily large.
Finally, we highlight another open problem: whether these games are always determined.