TITLE: A two player game on compact topological spaces
QUESTION [7 upvotes]: I've though of an infinite game that two players may play on a given topological space $(X,\tau)$. It goes like this. 
On turn $n$ Player I selects a point $x_n\in X$ and Player II selects a neighborhood $U_n$ of $x$. 
They play $\omega$ times, and Player I wins if $\{U_n\}_n$ is a cover of $X$. 
Thinking about it I've come up with what I believe are some true statements regarding this game. 
-If $(X,\tau)$ is not Lindelöf then Player II has always a winning strategy.
-If $(X,\tau)$ is countable than Player I has always a winning strategy.
-If $X=[0,1]$ then Player II has a winning strategy (just choose interval neighborhoods with increasingly smaller diameter, in such a way that the series given by adding them converges to a value smaller that $1$).
-If $X=\omega_1+1$ (or any successor ordinal) then Player I has a winning strategy. 
Now I think it might be that, given that $(X,\tau)$ is compact, one of the players always has a winning strategy (i.e. the game is determined), because open games are determined. I mean, this depends on whether open games on game spaces of the form $S^\omega$, not necessarily the Baire space, are always determined. 

My question is whether it is known if this dividing line among compact topologies, namely whether Player I or Player II has a winning strategy in this game, has strong consequences in any direction. 

For example it might be that this can be framed in terms of classical topological properties.

REPLY [3 votes]: According to the Wikipedia article on topological games, your game is known as the point-open game. Wikipedia has no more information about this game, but refers to the paper Rastislav Telgârsky, Topological games: on the 50th anniversary of the Banach–Mazur game, Rocky Mountain J. Math. 17 (1987), 227–276, which has a lot of information about it.