# Characterization of the Scheepers property by Scheepers game

$$\Omega$$: The collection of all $$\omega$$-covers of a space $$X$$. An open cover $$\mathcal U$$ of $$X$$ is said to be $$\omega$$-cover if $$X\notin\mathcal U$$ and for each finite $$F\subseteq X$$ there exists a $$U\in\mathcal U$$ such that $$F\subseteq U$$.

1. A space $$X$$ is said to have the Menger property if for each sequence $$(\mathcal U_n)$$ of open covers of $$X$$ there exists a sequence $$(\mathcal V_n)$$ such that for each $$n$$ $$\mathcal V_n$$ is a finite subset of $$\mathcal U_n$$ and $$\{\cup\mathcal V_n : n\in\mathbb N\}$$ covers $$X$$.

2. A space $$X$$ is said to have the Scheepers property if for each sequence $$(\mathcal U_n)$$ of open covers of $$X$$ there exists a sequence $$(\mathcal V_n)$$ such that for each $$n$$ $$\mathcal V_n$$ is a finite subset of $$\mathcal U_n$$ and $$\{\cup\mathcal V_n : n\in\mathbb N\}\in\Omega$$ or $$\cup\mathcal V_n=X$$ for some $$n$$.

The infinitely long games correponding to the Menger and Scheepers properties are defined as follows.

1. The Menger game on $$X$$ is played as follows. Players ONE and TWO play an inning for each positive integer $$n$$. In the $$n$$th inning ONE chooses an open cover $$\mathcal U_n$$ of $$X$$ and TWO responds by selecting a finite subset $$\mathcal V_n$$ of $$\mathcal U_n$$. TWO wins the play $$\mathcal U_1,\mathcal V_1,\dotsc,\mathcal U_n,\mathcal V_n,\dotsc$$ if $$\{\cup\mathcal V_n : n\in\mathbb N\}$$ covers $$X$$; otherwise ONE wins.

2. The Scheepers game on $$X$$ is played as follows. Players ONE and TWO play an inning for each positive integer $$n$$. In the $$n$$th inning ONE chooses an open cover $$\mathcal U_n$$ of $$X$$ and TWO responds by selecting a finite subset $$\mathcal V_n$$ of $$\mathcal U_n$$. TWO wins the play $$\mathcal U_1,\mathcal V_1,\dotsc,\mathcal U_n,\mathcal V_n,\dotsc$$ if $$\{\cup\mathcal V_n : n\in\mathbb N\}\in\Omega$$ or $$\cup\mathcal V_n=X$$ for some $$n$$; otherwise ONE wins.

It is well known that a space $$X$$ has the Menger property if and only if ONE does not have a winning strategy in the Menger game on $$X$$ (see here). We try to prove a similar result for the Scheepers game, but we fail. Does a similar result hold for the Scheepers game? 