If non-empty player has a winning strategy in Banach-Mazur game BM(X), then it also has in BM(Y)?

Question

Let $f:X\rightarrow Y$ be a continuous, open, surjection function and second player (non-empty) has a winning strategy (not important which one, say for simplicity stationary st.) in $BM(X)$. Then can we say the player has the same strategy in $BM(Y)$ ?

My attempt: 1) $\sigma_Y(U)=f(\sigma_X(f^{-1}(U)))$ while $\sigma_i$ is stationery st. in $BM(i)$

2) Trying to find a continuous, open, surjection function from a scattered space to rational numbers.

Some definition right here https://dantopology.wordpress.com/2012/06/08/the-banach-mazur-game/

Thanks for any help.