Γ-space

In mathematics, a **γ

{\displaystyle \gamma }

-space** (gamma space) is a topological space that satisfies a certain basic selection principle. An infinite cover of a topological space is an

    ω
  

{\displaystyle \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

    γ
  

{\displaystyle \gamma }

-cover if every point of this space belongs to all but finitely many members of this cover. A **γ

{\displaystyle \gamma }

-space** is a space in which every open

    ω
  

{\displaystyle \omega }

-cover contains a

    γ
  

{\displaystyle \gamma }

-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

    [
    
      N
    
    
      ]
      
        ∞
      
    
  

{\displaystyle [\mathbb {N} ]^{\infty }}

be the set of all infinite subsets of the set of natural numbers. A set

    A
    ⊂
    [
    
      N
    
    
      ]
      
        ∞
      
    
  

{\displaystyle A\subset [\mathbb {N} ]^{\infty }}

is centered if the intersection of finitely many elements of

    A
  

{\displaystyle A}

is infinite. Every set

    a
    ∈
    [
    
      N
    
    
      ]
      
        ∞
      
    
  

{\displaystyle a\in [\mathbb {N} ]^{\infty }}

we identify with its increasing enumeration, and thus the set

    a
  

{\displaystyle a}

we can treat as a member of the Baire space

        N
      
      
        
          N
        
      
    
  

{\displaystyle \mathbb {N} ^{\mathbb {N} }}

. Therefore,

    [
    
      N
    
    
      ]
      
        ∞
      
    
  

{\displaystyle [\mathbb {N} ]^{\infty }}

is a topological space as a subspace of the Baire space

        N
      
      
        
          N
        
      
    
  

{\displaystyle \mathbb {N} ^{\mathbb {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
    
    
      ]
      
        ∞
      
    
  

{\displaystyle [\mathbb {N} ]^{\infty }}

that is centered has a pseudointersection.

Let

    X
  

{\displaystyle X}

be a topological space. The

    γ
  

{\displaystyle \gamma }

-has a pseudo intersection if there is a set game played on

    X
  

{\displaystyle X}

is a game with two players Alice and Bob.

1st round: Alice chooses an open

    ω
  

{\displaystyle \omega }

-cover

          U
        
      
      
        1
      
    
  

{\displaystyle {\mathcal {U}}_{1}}

of

    X
  

{\displaystyle X}

. Bob chooses a set

      U
      
        1
      
    
    ∈
    
      
        
          U
        
      
      
        1
      
    
  

{\displaystyle U_{1}\in {\mathcal {U}}_{1}}

.

2nd round: Alice chooses an open

    ω
  

{\displaystyle \omega }

-cover

          U
        
      
      
        2
      
    
  

{\displaystyle {\mathcal {U}}_{2}}

of

    X
  

{\displaystyle X}

. Bob chooses a set

      U
      
        2
      
    
    ∈
    
      
        
          U
        
      
      
        2
      
    
  

{\displaystyle U_{2}\in {\mathcal {U}}_{2}}

.

etc.

If

    {
    
      U
      
        n
      
    
    :
    n
    ∈
    
      N
    
    }
  

{\displaystyle \{U_{n}:n\in \mathbb {N} \}}

is a

    γ
  

{\displaystyle \gamma }

-cover of the space

    X
  

{\displaystyle 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

    γ
  

{\displaystyle \gamma }

-space iff Alice has no winning strategy in the

    γ
  

{\displaystyle \gamma }

-game played on this space.

• A topological space is a γ-space if and only if it satisfies

        S
      
      
        1
      
    
    (
    Ω
    ,
    Γ
    )
  

  {\displaystyle {\text{S}}_{1}(\Omega ,\Gamma )}

selection principle.

• Every Lindelöf space of cardinality less than the pseudointersection number

        p
  

  {\displaystyle {\mathfrak {p}}}

is a

    γ
  

{\displaystyle \gamma }

-space.

• Every

    γ
  

  {\displaystyle \gamma }

-space is a Rothberger space, and thus it has strong measure zero.

• Let

    X
  

  {\displaystyle X}

be a Tychonoff space, and

    C
    (
    X
    )
  

{\displaystyle C(X)}

be the space of continuous functions

    f
    :
    X
    →
    
      R
    
  

{\displaystyle f\colon X\to \mathbb {R} }

with pointwise convergence topology. The space

    X
  

{\displaystyle X}

is a

    γ
  

{\displaystyle \gamma }

-space if and only if

    C
    (
    X
    )
  

{\displaystyle C(X)}

is Fréchet–Urysohn if and only if

    C
    (
    X
    )
  

{\displaystyle C(X)}

is strong Fréchet–Urysohn.

• Let

    A
  

  {\displaystyle A}

be a

          (
        
        
          
            Ω
          
          
            Γ
          
        
        
          )
        
      
    
  

{\displaystyle {\binom {\mathbf {\Omega } }{\mathbf {\Gamma } }}}

subset of the real line, and

    M
  

{\displaystyle M}

be a meager subset of the real line. Then the set

    A
    +
    M
    =
    {
    a
    +
    x
    :
    a
    ∈
    A
    ,
    x
    ∈
    M
    }
  

{\displaystyle A+M=\{a+x:a\in A,x\in M\}}

is meager.

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