γ-space

In mathematics, a $\gamma$-space (gamma space) is a topological space that satisfies a certain 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 $[\mathbb{N}{]}^{\mathrm{\infty }}$ be the set of all infinite subsets of the set of natural numbers. A set $A\subset [\mathbb{N}{]}^{\mathrm{\infty }}$ is centered if the intersection of finitely many elements of $A$ is infinite. Every set $a\in [\mathbb{N}{]}^{\mathrm{\infty }}$ we identify with its increasing enumeration, and thus the set $a$ we can treat as a member of the Baire space ${\mathbb{N}}^{\mathbb{N}}$. Therefore, $[\mathbb{N}{]}^{\mathrm{\infty }}$ is a topological space as a subspace of the Baire space ${\mathbb{N}}^{\mathbb{N}}$. A zero-dimensional separable metric space is a γ-space if and only if every continuous image of that space into the space $[\mathbb{N}{]}^{\mathrm{\infty }}$ that 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 ${𝒰}_1$ of $X$. Bob chooses a set $U_1\in {𝒰}_1$.

2nd round: Alice chooses an open $\omega$-cover ${𝒰}_2$ of $X$. Bob chooses a set $U_2\in {𝒰}_2$.

etc.

If $\{U_n:n\in \mathbb{N}\}$ 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 ${\text{S}}_1(\mathrm{\Omega },\mathrm{\Gamma })$ selection principle.
• Every Lindelöf space of cardinality less than the pseudointersection number $𝔭$ 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 $f:X\to ℝ$ 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 $(\genfrac{}{}{0ex}{}{\mathrm{\Omega }}{\mathrm{\Gamma }})$ subset of the real line, and $M$ be a meager subset of the real line. Then the set $A+M=\{a+x:a\in A,x\in M\}$ is meager.