γ-space
   HOME

TheInfoList



OR:

In mathematics, a \gamma-space (gamma space) is a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
that satisfies a certain basic
selection principle In mathematics, a selection principle is a rule asserting the possibility of obtaining mathematically significant objects by selecting elements from given sequences of Set (mathematics), sets. The theory of selection principles studies these pri ...
. 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. 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
mathbb Blackboard bold is a style of writing Emphasis (typography), bold symbols on a blackboard by doubling certain strokes, commonly used in mathematical lectures, and the derived style of typeface used in printed mathematical texts. The style is most ...
\infty be the set of all infinite subsets of the set of natural numbers. A set A\subset
mathbb Blackboard bold is a style of writing Emphasis (typography), bold symbols on a blackboard by doubling certain strokes, commonly used in mathematical lectures, and the derived style of typeface used in printed mathematical texts. The style is most ...
\infty is centered if the intersection of finitely many elements of A is infinite. Every set a\in
mathbb Blackboard bold is a style of writing Emphasis (typography), bold symbols on a blackboard by doubling certain strokes, commonly used in mathematical lectures, and the derived style of typeface used in printed mathematical texts. The style is most ...
\infty we identify with its increasing enumeration, and thus the set a we can treat as a member of the
Baire space In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are ...
\mathbb^\mathbb. Therefore,
mathbb Blackboard bold is a style of writing Emphasis (typography), bold symbols on a blackboard by doubling certain strokes, commonly used in mathematical lectures, and the derived style of typeface used in printed mathematical texts. The style is most ...
\infty is a topological space as a subspace of the Baire space \mathbb^\mathbb. A
zero-dimensional In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space. A graphical ...
separable
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
is a γ-space if and only if every continuous image of that space into the space
mathbb Blackboard bold is a style of writing Emphasis (typography), bold symbols on a blackboard by doubling certain strokes, commonly used in mathematical lectures, and the derived style of typeface used in printed mathematical texts. The style is most ...
\infty that is centered has a pseudointersection.


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 \mathcal_1 of X. Bob chooses a set U_1\in \mathcal_1. 2nd round: Alice chooses an open \omega-cover \mathcal_2 of X. Bob chooses a set U_2\in \mathcal_2. etc. If \ 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 In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
is a γ-space if and only if it satisfies \text_1(\Omega,\Gamma)
selection principle In mathematics, a selection principle is a rule asserting the possibility of obtaining mathematically significant objects by selecting elements from given sequences of Set (mathematics), sets. The theory of selection principles studies these pri ...
. * Every
Lindelöf space In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of ''compactness'', which requires the existence of a ''finite'' sub ...
of cardinality less than the pseudointersection number \mathfrak is a \gamma-space. * Every \gamma-space is a Rothberger space, and thus it has strong measure zero. * Let X be a
Tychonoff space In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space is any completely regular space that is also a ...
, and C(X) be the space of continuous functions f\colon X\to\mathbb with
pointwise convergence In mathematics, pointwise convergence is one of Modes of convergence (annotated index), various senses in which a sequence of function (mathematics), functions can Limit (mathematics), converge to a particular function. It is weaker than uniform co ...
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 subset of the real line, and M be a meager subset of the real line. Then the set A+M=\ is meager.


References

{{Reflist General topology