View source for Γ-space

{{Short description|Topological space}}
{{DISPLAYTITLE:γ-space}}
In mathematics, a '''<math>\gamma</math>-space''' (gamma space) is a [[topological space]] that satisfies a certain basic [[selection principle]]. 
An infinite cover of a topological space is an <math>\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>\gamma</math>-cover if every point of this space belongs to all but finitely many members of this cover.
A '''<math>\gamma</math>-space''' is a space in which every open <math>\omega</math>-cover contains a <math>\gamma</math>-cover.

== History ==
Gerlits and Nagy introduced the notion of γ-spaces.<ref name=":0">{{cite journal|last1=Gerlits|first1=J.|last2=Nagy|first2=Zs.|title=Some properties of <math>C(X)</math>, I|
journal=Topology and Its Applications|date=1982|volume=14|issue=2|pages=151–161|doi=10.1016/0166-8641(82)90065-7|doi-access=free}}</ref> 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>[\mathbb{N}]^\infty</math> be the set of all infinite subsets of the set of natural numbers. A set <math>A\subset [\mathbb{N}]^\infty</math> is centered if the intersection of finitely many elements of <math>A</math> is infinite. Every set <math>a\in[\mathbb{N}]^\infty</math> we identify with its increasing enumeration, and thus the set <math>a</math> we can treat as a member of the [[Baire space]] <math>\mathbb{N}^\mathbb{N}</math>. Therefore, <math>[\mathbb{N}]^\infty</math> is a topological space as a subspace of the Baire space <math>\mathbb{N}^\mathbb{N}</math>. A [[Zero-dimensional space|zero-dimensional]] [[separable space|separable]] [[metric space]] is a γ-space if and only if every continuous image of that space into the space <math>[\mathbb{N}]^\infty</math> that is centered has a [[Pseudo-intersection|pseudointersection]].<ref name="Reclaw">{{Cite journal|last=Recław|first=Ireneusz|year=1994|title=Every Lusin set is undetermined in the point-open game|url=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|journal=Fundamenta Mathematicae|volume=144|pages=43–54|doi=10.4064/fm-144-1-43-54|doi-access=free}}</ref>

=== Topological game characterization ===
Let <math>X</math> be a topological space. The <math>\gamma</math>-has a pseudo intersection if there is a set game played on <math>X</math> is a game with two players Alice and Bob.

'''1st round''': Alice chooses an open <math>\omega</math>-cover <math>\mathcal{U}_1</math> of <math>X</math>. Bob chooses a set <math>U_1\in \mathcal{U}_1</math>.

'''2nd round''': Alice chooses an open <math>\omega</math>-cover <math>\mathcal{U}_2</math> of <math>X</math>. Bob chooses a set <math>U_2\in \mathcal{U}_2</math>.

'''etc.'''

If <math>\{U_n:n\in\mathbb{N}\}</math> is a <math>\gamma</math>-cover of the space <math>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>\gamma</math>-space iff Alice has no winning strategy in the <math>\gamma</math>-game played on this space.<ref name=":0" />

== Properties ==
* A [[topological space]] is a γ-space if and only if it satisfies <math>\text{S}_1(\Omega,\Gamma)</math> [[selection principle]].<ref name=":0" />
* Every [[Lindelöf space]] of cardinality less than the [[Pseudo-intersection|pseudointersection]] number <math>\mathfrak{p}</math> is a <math>\gamma</math>-space.
* Every <math>\gamma</math>-space is a [[Rothberger space]],<ref name=coc1>{{cite journal|last1=Scheepers|first1=Marion|
title=Combinatorics of open covers I: Ramsey theory|
journal=Topology and Its Applications|date=1996|volume=69|pages=31–62|doi=10.1016/0166-8641(95)00067-4|url=https://scholarworks.boisestate.edu/math_facpubs/90|doi-access=free}}</ref>  and thus it has [[Strong measure zero set|strong measure zero]].

* Let <math>X</math> be a  [[Tychonoff space]], and <math>C(X)</math> be  the space of continuous functions <math>f\colon X\to\mathbb{R}</math> with [[pointwise convergence]] topology. The space <math>X</math> is a <math>\gamma</math>-space if and only if <math>C(X)</math> is [[Fréchet–Urysohn space|Fréchet–Urysohn]]  if and only if <math>C(X)</math> is [[Fréchet–Urysohn space|strong Fréchet–Urysohn]].<ref name=":0" />
* Let <math>A</math> be a <math>\binom{\mathbf{\Omega}}{\mathbf{\Gamma}}</math> subset of the real line, and <math>M</math> be a [[meagre set|meager]] subset of the real line. Then the set <math>A+M=\{a+x:a\in A, x\in M\}</math> is meager.<ref>{{cite journal|last1=Galvin|first1=Fred|last2=Miller|first2=Arnold|title=<math>\gamma</math>-sets and other singular sets of real numbers|
journal=Topology and Its Applications|date=1984|volume=17|issue=2|pages=145–155|doi=10.1016/0166-8641(84)90038-5|doi-access=free}}</ref>

== References ==
{{Reflist}}

[[Category:General topology]]
[[Category:Topological spaces| ]]