γ-space

Let ${\displaystyle [\mathbb {N} ]^{\infty }}$ be the set of all infinite subsets of the set of natural numbers. A set ${\displaystyle A\subset [\mathbb {N} ]^{\infty }}$is centered if the intersection of finitely many elements of ${\displaystyle A}$ is infinite. Every set ${\displaystyle a\in [\mathbb {N} ]^{\infty }}$we identify with its increasing enumeration, and thus the set ${\displaystyle a}$ we can treat as a member of the Baire space ${\displaystyle \mathbb {N} ^{\mathbb {N} }}$. Therefore, ${\displaystyle [\mathbb {N} ]^{\infty }}$is a topological space as a subspace of the Baire space ${\displaystyle \mathbb {N} ^{\mathbb {N} }}$. A zero-dimensional separable metric space is a γ-space if and only if every continuous image of that space into the space ${\displaystyle [\mathbb {N} ]^{\infty }}$that is centered has a pseudointersection.[2]

Let ${\displaystyle X}$ be a topological space. The ${\displaystyle \gamma }$-has a pseudo intersection if there is a set game played on ${\displaystyle X}$ is a game with two players Alice and Bob.

1st round: Alice chooses an open ${\displaystyle \omega }$-cover ${\displaystyle {\mathcal {U}}_{1}}$ of ${\displaystyle X}$. Bob chooses a set ${\displaystyle U_{1}\in {\mathcal {U}}_{1}}$.

2nd round: Alice chooses an open ${\displaystyle \omega }$-cover ${\displaystyle {\mathcal {U}}_{2}}$ of ${\displaystyle X}$. Bob chooses a set ${\displaystyle U_{2}\in {\mathcal {U}}_{2}}$.

etc.

If ${\displaystyle \{U_{n}:n\in \mathbb {N} \}}$ is a ${\displaystyle \gamma }$-cover of the space ${\displaystyle X}$, then Bob wins the game. Otherwise, Alice wins.

A player has a winning strategy if he knows how to play in order to win the game (formally, a winning strategy is a function).

A topological space is a ${\displaystyle \gamma }$-space iff Alice has no winning strategy in the ${\displaystyle \gamma }$-game played on this space.[1]