# Rothberger game and Meager set.

Someone know examples of topological spaces of first category and in which Player II has a winning strategy in the Rothberger game?

Remember that:

The Rothberger game on a topological space $$X$$ is played according to the following rules:

In each inning $$n\in\omega$$, Player I chooses an open cover $$\mathcal{U_n}$$ of $$X$$, and then Player II picks an open set $$U_{n}\in\mathcal{U}_{n}$$. At the end of the play $$\langle \mathcal{U}_{0}, U_{0}, \mathcal{U}_{1}, U_{1}, ..., \mathcal{U}_{n}, U_{n}, .... \rangle$$. The winner is Player II if $$X\subseteq\bigcup_{n\in\omega}U_n$$, and Player I otherwise.

Thanks

Again, it seems the right order topology on $$\Bbb R$$ works. Meagerness is verified here. The basis elements of this topology are $$B_y=\{x\in\Bbb R\mid x>y\}$$. At the first stage, choose an open set $$U_1$$ containing $$-1$$. It must be either an open ray $$B_y$$ such that $$y<-1$$, or $$U_1=\Bbb R$$. In the latter case, we've won. At the $$n$$-th stage, choose $$U_n$$ containing $$-n$$, the same reasoning goes through. It's easy to see $$\Bbb R= \bigcup_{n\in\omega}\{x\in \Bbb R\mid x>-n\}\subset\bigcup_{n\in \omega}U_n$$.
• @user987 Actually, $\Bbb Q$ is meager and countable. And every countable space is Rothberger. (if you don't care countability) – YuiTo Cheng Apr 24 '19 at 3:50