Let $X$ be a non-empty topological space. The Banach-Mazur game on $X$, $\textsf{BM}(X)$, is played as follows: Players I and II play an inning per positive integer. In the $n$-th inning Player I chooses a nonempty open set $A_n$; Player II responds with a nonempty open set $B_n \subseteq A_n$. Player I must also obey the rule that for each $n$, $A_{n+1} \subseteq B_n$. A play $A_1, B_1, \cdots A_n, B_n$ is won by Player II if $\bigcap_{n\in\omega}B_n \not=\emptyset$; otherwise, Player I wins.
Some facts about the Banach-Mazur game:
- A nonempty topological space $X$ is a Baire space if and only if Player I has no winning strategy in the Banach-Mazur game $\textsf{BM}(X)$.
- Let $X$ be a topological space where Player II has a winning strategy in the $\textsf{BM}(X)$ and let $Y$ a Baire space then $X\times Y$ is a Baire space. In other words II $\uparrow \textsf{BM}(X)$ implies $X$ productively Baire.
- Every $(X,d)$ metric space without isolated points where Player II has a winning strategy in $\textsf{BM}(X)$ contains a subspace homeomorphic to the Cantor set.
We say that a topological space $X$ is undetermined if the Banach-Mazur game is undetermined when is played in $X$, that's, neither player has a winning strategy in $\textsf{BM}(X)$.
Some examples of undetermined spaces :
- Let $B\subseteq \mathbb{R}$ be a Bernstein set, then $B$ is a undetermined space. (It follows from the fact that Bernstein sets are Baire spaces and from 3.)
- There are Baire spaces which are not productively Baire, in particular this spaces are undetermined. (It follows from 2.)
My question is the following: Anyone know more examples of undetermined spaces in the real line?
Thanks
