In mathematics, a -space is a topological space that satisfies a certain a basic selection principle. An infinite cover of a topological space is an -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 -cover if every point of this space belongs to all but finitely many members of this cover. A -space is a space in which for every open -cover contains a -cover.
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.
Let be the set of all infinite subsets of the set of natural numbers. A set is centered if the intersection of finitely many elements of is infinite. Every set we identify with its increasing enumeration, and thus the set we can treat as a member of the Baire space . Therefore, is a topological space as a subspace of the Baire space . A zero-dimensional separable metric space is a γ-space if and only if every continuous image of that space into the space that is centered has a pseudointersection.
Topological game characterization
Let be a topological space. The -has a pseudo intersection if there is a set game played on is a game with two players Alice and Bob.
1st round: Alice chooses an open -cover of . Bob chooses a set .
2nd round: Alice chooses an open -cover of . Bob chooses a set .
If is a -cover of the space , 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 γ-space if and only if it satisfies selection principle.
- Every Lindelöf space of cardinality less than the pseudointersection number is a -space.
- Every -space is a Rothberger space, and thus it has strong measure zero.
- Let be a Tychonoff space, and be the space of continuous functions with pointwise convergence topology. The space is a -space if and only if is Fréchet–Urysohn if and only if is strong Fréchet–Urysohn.
- Let be a subset of the real line, and be a meager subset of the real line. Then the set is meager.
- Gerlits, J.; Nagy, Zs. (1982). "Some properties of , 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.
- 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.
- Galvin, Fred; Miller, Arnold (1984). "-sets and other singular sets of real numbers". Topology and Its Applications. 17 (2): 145–155. doi:10.1016/0166-8641(84)90038-5.