γ-space

In mathematics, a $\gamma$-space is a topological space that satisfies a certain a basic selection principle.
An infinite cover of a topological space is an $\omega$-cover if every finite subset of this space is contained in some member of the cover, and the whole space is not a member the cover. A cover of a topological space is a $\gamma$-cover if every point of this space belongs to all but finitely many members of this cover.
A $\gamma$-space is a space in which every open $\omega$-cover contains a $\gamma$-cover.

History

Gerlits and Nagy introduced the notion of γ-spaces. They listed some topological properties and enumerated them by Greek letters. The above property was the third one on this list, and therefore it is called the γ-property.

Characterizations

Combinatorial characterization

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

Topological game characterization

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

1st round: Alice chooses an open $\omega$-cover $\mathcal_1$ of $X$. Bob chooses a set $U_1\in \mathcal_1$.

2nd round: Alice chooses an open $\omega$-cover $\mathcal_2$ of $X$. Bob chooses a set $U_2\in \mathcal_2$.

etc.

If $\$ is a $\gamma$-cover of the space $X$, 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 $\gamma$-space iff Alice has no winning strategy in the $\gamma$-game played on this space.

Properties

* A topological space is a γ-space if and only if it satisfies $\text_1(\Omega,\Gamma)$ selection principle.
* Every Lindelöf space of cardinality less than the pseudointersection number $\mathfrak$ is a $\gamma$-space.
* Every $\gamma$-space is a Rothberger space, and thus it has strong measure zero.

* Let $X$ be a Tychonoff space, and $C(X)$ be the space of continuous functions $f\colon X\to\mathbb$ with pointwise convergence topology. The space $X$ is a $\gamma$-space if and only if $C(X)$ is Fréchet–Urysohn if and only if $C(X)$ is strong Fréchet–Urysohn.
* Let $A$ be a $\binom$ subset of the real line, and $M$ be a meager subset of the real line. Then the set $A+M=\$ is meager.

References

{{Reflist

General topology