9 problems

Telgársky's conjecture. For every $k\geq 2$, there is a space $X_k$ such that player ${\scriptsize\operatorname{NONEMPTY}}$ has a winning $(k+1)$-tactic in the Banach–Mazur game on…

Let $\kappa$ be an infinite cardinal. In the game $SMG([\kappa]^{<\aleph_{0}})$, player ONE plays an increasing sequence of members of the generated ideal and TWO responds with mem…

Non-determinacy problem. The game $G_{\omega}(Y,\mathbb{A})$ is not determined.

Let $X$ be a topological space, and let $\mathcal F[X]$ denote its finite-set hyperspace. A space is discretely selective if it has the selection property referred to in the preced…

Let $X$ be a $T_1$ space, and let $G_c(\mathcal O,\mathcal O)$ denote the compact-open selection game in which ONE and TWO choose open covers and TWO selects finite subfamilies. Wr…

Winning-strategy conjecture. For each positive integer $n$, ONE has a winning strategy in

Let $X$ be a topological space, let $\pi(X)$ be the least cardinality of a $\pi$-base for $X$, and let $\operatorname{BM}(X)$ denote the Banach–Mazur game on $X$. For positive card…

Let $X$ be a topological space. The strongly star-Hurewicz property means that for every sequence $(\mathcal U_n:n\in\mathbb N)$ of open covers of $X$, there are finite sets…

Consider a Sprouts starting position with $n$ spots on an orientable compact surface, and let its nimber be the least excluded nonnegative integer among the nimbers of the position…