Γ-space

In mathematics, a ${\displaystyle \gamma }$-space (gamma space) is a topological space that satisfies a certain basic selection principle. An infinite cover of a topological space is an ${\displaystyle \omega }$-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 ${\displaystyle \gamma }$-cover if every point of this space belongs to all but finitely many members of this cover. A ${\displaystyle \gamma }$-space is a space in which every open ${\displaystyle \omega }$-cover contains a ${\displaystyle \gamma }$-cover.

History

Gerlits and Nagy introduced the notion of γ-spaces.[1] 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 ${\displaystyle [\mathbb {N} ]^{\infty }}$ 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]

Topological game characterization

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]

Properties

• A topological space is a γ-space if and only if it satisfies ${\displaystyle {\text{S}}_{1}(\Omega ,\Gamma )}$ selection principle.[1]
• Every Lindelöf space of cardinality less than the pseudointersection number ${\displaystyle {\mathfrak {p}}}$ is a ${\displaystyle \gamma }$-space.
• Every ${\displaystyle \gamma }$-space is a Rothberger space,[3] and thus it has strong measure zero.

• Let ${\displaystyle X}$ be a Tychonoff space, and ${\displaystyle C(X)}$ be the space of continuous functions ${\displaystyle f\colon X\to \mathbb {R} }$ with pointwise convergence topology. The space ${\displaystyle X}$ is a ${\displaystyle \gamma }$-space if and only if ${\displaystyle C(X)}$ is Fréchet–Urysohn if and only if ${\displaystyle C(X)}$ is strong Fréchet–Urysohn.[1]
• Let ${\displaystyle A}$ be a ${\displaystyle {\binom {\mathbf {\Omega } }{\mathbf {\Gamma } }}}$ subset of the real line, and ${\displaystyle M}$ be a meager subset of the real line. Then the set ${\displaystyle A+M=\{a+x:a\in A,x\in M\}}$ is meager.[4]

References

1. Gerlits, J.; Nagy, Zs. (1982). "Some properties of ${\displaystyle C(X)}$, I". Topology and Its Applications. 14 (2): 151–161. doi:10.1016/0166-8641(82)90065-7.
2. Recław, Ireneusz (1994). "Every Lusin set is undetermined in the point-open game". Fundamenta Mathematicae. 144: 43–54. doi:10.4064/fm-144-1-43-54.
3. Scheepers, Marion (1996). "Combinatorics of open covers I: Ramsey theory". Topology and Its Applications. 69: 31–62. doi:10.1016/0166-8641(95)00067-4.
4. Galvin, Fred; Miller, Arnold (1984). "${\displaystyle \gamma }$-sets and other singular sets of real numbers". Topology and Its Applications. 17 (2): 145–155. doi:10.1016/0166-8641(84)90038-5.