Characterization of the Scheepers property by Scheepers game

Question

$\Omega$: 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?

Answer 1

The theorem you are looking for is a special case of Theorem 6 of the paper Partition relations for Hurewicz-type selection hypotheses. The needed definitions in the following general theorem are provided in the linked paper.

The statement is slightly different, but the proof shows the equivalence to what you look for (such equivalences are routine, in any case).