γ-space

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.

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 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.[2]

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 .

etc.

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 -space iff Alice has no winning strategy in the -game played on this space.[1]

Properties

  • 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.[1]
  • Let be a subset of the real line, and be a meager subset of the real line. Then the set is meager.[4]

References

  1. 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.
  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). "-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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