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David Gale's subset take-away game is a game where two players A and B play with a finite set $S$. Players alternately choose proper subsets of $S$ such that if a subset is chosen, then none of its subsets may be chosen after that. Gale conjectured that the second player always wins; this has been checked for $|S|\leq 6$.

Make the following modification to the game: specify a topology $\tau$ on $S$. Two players A and B play with $(S,\tau)$. Players alternately choose proper open subsets of $S$ such that if an open subset is chosen, then none of its subsets may be chosen after that.

Question: For what nontrivial topologies on $S$ does the second player always win?

David Gale's subset take-away game is a game where two players A and B play with a finite set $S$. Players alternately choose proper nonempty subsets of $S$ such that if a subset is chosen, then none of its subsets may be chosen after that. Gale conjectured that the second player always wins; this has been checked for $|S|\leq 6$.

Make the following modification to the game: specify a topology $\tau$ on $S$. Two players A and B play with $(S,\tau)$. Players alternately choose proper nonempty open subsets of $S$ such that if an open subset is chosen, then none of its subsets may be chosen after that.

Question: For what nontrivial topologies on $S$ does the second player always win?

David Gale's subset take-away game is a game where two players A and B play with a finite set $S$. Players alternately choose subsets of $S$ such that if a subset is chosen, then none of its subsets may be chosen after that. Gale conjectured that the second player always wins; this has been checked for $|S|\leq 6$.

Make the following modification to the game: specify a topology $\tau$ on $S$. Two players A and B play with $(S,\tau)$. Players alternately choose open subsets of $S$ such that if an open subset is chosen, then none of its subsets may be chosen after that.

Question: Does the second player always win?

David Gale's subset take-away game is a game where two players A and B play with a finite set $S$. Players alternately choose proper subsets of $S$ such that if a subset is chosen, then none of its subsets may be chosen after that. Gale conjectured that the second player always wins; this has been checked for $|S|\leq 6$.

Make the following modification to the game: specify a topology $\tau$ on $S$. Two players A and B play with $(S,\tau)$. Players alternately choose proper open subsets of $S$ such that if an open subset is chosen, then none of its subsets may be chosen after that.

Question: For what nontrivial topologies on $S$ does the second player always win?