Two players play the following game. Before the beginning $"0."$ is written on the board. The first player writes any (finite) sequence of digits. The second one then writes only one digit. Then the players take turns. The first player wins if the number on the board is of repeating decimal type. The question is, which of players has a winning strategy.
My guess is that the first player can win despite the turns of the second player. I tried to combine facts that for any $m$ set of numbers $0.1a_1a_2\dots a_m$ is open, and that repeating decimals are dense in $[0;1]$, but I can't explicitly show the winning strategy.
Any help would be appreciated. Thanks!
The second player will always have a winning strategy if the "target" set ($\mathbb{Q}$ in this case) is countable: If that set is $\{x_n\}_{n\in\mathbb{N}}$ then the second player only has to choose, on the $n^{th}$ round, a digit that makes the decimal expansion of the number constructed up to that point different from the decimal expansion of $x_n$. To avoid ambiguities, the digit can be limited to just two values, say 1 and 2 (so there's no issue with non unique decimal expansions)