Γ-space explained

In mathematics, a

\gamma -space is a topological space that satisfies a certain a 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.^[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

[ N ] ^{infty} be the set of all infinite subsets of the set of natural numbers. A set A\subset [ N ] ^{infty} is centered if the intersection of finitely many elements of A is infinite. Every set a\in[ N ] ^{infty} we identify with its increasing enumeration, and thus the set a we can treat as a member of the Baire space N ^{N} . Therefore, [ N ] ^{infty} is a topological space as a subspace of the Baire space N ^{N} . A zero-dimensional separable metric space is a γ-space if and only if every continuous image of that space into the space [ N ] ^{infty} that is centered has a pseudointersection.^[2]

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 l{U} _{1} of X . Bob chooses a set U _{1\in} l{U} _{1} .

2nd round: Alice chooses an open

\omega -cover l{U} _{2} of X . Bob chooses a set U _{2\in} l{U} _{2} .

etc.

If

\{U _{n:n\inN\}} 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

S _{1(\Omega,\Gamma)} selection principle.

• Every Lindelöf space of cardinality less than the pseudointersection number

ak{p} is a \gamma -space.

• Every

\gamma -space is a Rothberger space,^[3] and thus it has strong measure zero.

• Let

X be a Tychonoff space, and C(X) be the space of continuous functions f\colon 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 \binom{ \Omega } 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.^[4]

Notes and References

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