γ-space
In mathematics, a [math]\displaystyle{ \gamma }[/math]-space is a topological space that satisfies a certain a basic selection principle.
An infinite cover of a topological space is an [math]\displaystyle{ \omega }[/math]-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 [math]\displaystyle{ \gamma }[/math]-cover if every point of this space belongs to all but finitely many members of this cover.
A [math]\displaystyle{ \gamma }[/math]-space is a space in which every open [math]\displaystyle{ \omega }[/math]-cover contains a [math]\displaystyle{ \gamma }[/math]-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 [math]\displaystyle{ [\mathbb{N}]^\infty }[/math] be the set of all infinite subsets of the set of natural numbers. A set [math]\displaystyle{ A\subset [\mathbb{N}]^\infty }[/math]is centered if the intersection of finitely many elements of [math]\displaystyle{ A }[/math] is infinite. Every set [math]\displaystyle{ a\in[\mathbb{N}]^\infty }[/math]we identify with its increasing enumeration, and thus the set [math]\displaystyle{ a }[/math] we can treat as a member of the Baire space [math]\displaystyle{ \mathbb{N}^\mathbb{N} }[/math]. Therefore, [math]\displaystyle{ [\mathbb{N}]^\infty }[/math]is a topological space as a subspace of the Baire space [math]\displaystyle{ \mathbb{N}^\mathbb{N} }[/math]. A zero-dimensional separable metric space is a γ-space if and only if every continuous image of that space into the space [math]\displaystyle{ [\mathbb{N}]^\infty }[/math]that is centered has a pseudointersection.[2]
Topological game characterization
Let [math]\displaystyle{ X }[/math] be a topological space. The [math]\displaystyle{ \gamma }[/math]-has a pseudo intersection if there is a set game played on [math]\displaystyle{ X }[/math] is a game with two players Alice and Bob.
1st round: Alice chooses an open [math]\displaystyle{ \omega }[/math]-cover [math]\displaystyle{ \mathcal{U}_1 }[/math] of [math]\displaystyle{ X }[/math]. Bob chooses a set [math]\displaystyle{ U_1\in \mathcal{U}_1 }[/math].
2nd round: Alice chooses an open [math]\displaystyle{ \omega }[/math]-cover [math]\displaystyle{ \mathcal{U}_2 }[/math] of [math]\displaystyle{ X }[/math]. Bob chooses a set [math]\displaystyle{ U_2\in \mathcal{U}_2 }[/math].
etc.
If [math]\displaystyle{ \{U_n:n\in\mathbb{N}\} }[/math] is a [math]\displaystyle{ \gamma }[/math]-cover of the space [math]\displaystyle{ X }[/math], 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 [math]\displaystyle{ \gamma }[/math]-space iff Alice has no winning strategy in the [math]\displaystyle{ \gamma }[/math]-game played on this space.[1]
Properties
- A topological space is a γ-space if and only if it satisfies [math]\displaystyle{ \text{S}_1(\Omega,\Gamma) }[/math] selection principle.[1]
- Every Lindelöf space of cardinality less than the pseudointersection number [math]\displaystyle{ \mathfrak{p} }[/math] is a [math]\displaystyle{ \gamma }[/math]-space.
- Every [math]\displaystyle{ \gamma }[/math]-space is a Rothberger space,[3] and thus it has strong measure zero.
- Let [math]\displaystyle{ X }[/math] be a Tychonoff space, and [math]\displaystyle{ C(X) }[/math] be the space of continuous functions [math]\displaystyle{ f\colon X\to\mathbb{R} }[/math] with pointwise convergence topology. The space [math]\displaystyle{ X }[/math] is a [math]\displaystyle{ \gamma }[/math]-space if and only if [math]\displaystyle{ C(X) }[/math] is Fréchet–Urysohn if and only if [math]\displaystyle{ C(X) }[/math] is strong Fréchet–Urysohn.[1]
- Let [math]\displaystyle{ A }[/math] be a [math]\displaystyle{ \binom{\mathbf{\Omega}}{\mathbf{\Gamma}} }[/math] subset of the real line, and [math]\displaystyle{ M }[/math] be a meager subset of the real line. Then the set [math]\displaystyle{ A+M=\{a+x:a\in A, x\in M\} }[/math] is meager.[4]
References
- ↑ 1.0 1.1 1.2 1.3 Gerlits, J.; Nagy, Zs. (1982). "Some properties of [math]\displaystyle{ C(X) }[/math], I". Topology and Its Applications 14 (2): 151–161. doi:10.1016/0166-8641(82)90065-7.
- ↑ 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. http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-fmv144i1p43bwm?q=bwmeta1.element.bwnjournal-number-fm-1994-144-1;3&qt=CHILDREN-STATELESS.
- ↑ 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. https://scholarworks.boisestate.edu/math_facpubs/90.
- ↑ Galvin, Fred; Miller, Arnold (1984). "[math]\displaystyle{ \gamma }[/math]-sets and other singular sets of real numbers". Topology and Its Applications 17 (2): 145–155. doi:10.1016/0166-8641(84)90038-5.
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