Projective hierarchy

<p>In the mathematical field of <a href="/facts/Descriptive_set_theory/ek0tvUXb">descriptive set theory</a>, a subset 
  
    
      
        A
      
    
    {\displaystyle A}
  
 of a <a href="/facts/Polish_space/CkUFcGap">Polish space</a> 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is projective if it is 
  
    
      
        
          
            Σ
          
          
            n
          
          
            1
          
        
      
    
    {\displaystyle {\boldsymbol {\Sigma }}_{n}^{1}}
  
 for some positive integer 
  
    
      
        n
      
    
    {\displaystyle n}
  
.  Here 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is
</p>
<ul><li>
  
    
      
        
          
            Σ
          
          
            1
          
          
            1
          
        
      
    
    {\displaystyle {\boldsymbol {\Sigma }}_{1}^{1}}
  
 if 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is <a href="/facts/Analytic_set/CJn9ddXT">analytic</a></li>
<li>
  
    
      
        
          
            Π
          
          
            n
          
          
            1
          
        
      
    
    {\displaystyle {\boldsymbol {\Pi }}_{n}^{1}}
  
 if the <a href="/facts/Complement_(set_theory)/yWUePIYY">complement</a> of 
  
    
      
        A
      
    
    {\displaystyle A}
  
, 
  
    
      
        X
        ∖
        A
      
    
    {\displaystyle X\setminus A}
  
, is 
  
    
      
        
          
            Σ
          
          
            n
          
          
            1
          
        
      
    
    {\displaystyle {\boldsymbol {\Sigma }}_{n}^{1}}
  
</li>
<li>
  
    
      
        
          
            Σ
          
          
            n
            +
            1
          
          
            1
          
        
      
    
    {\displaystyle {\boldsymbol {\Sigma }}_{n+1}^{1}}
  
 if there is a Polish space 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 and a 
  
    
      
        
          
            Π
          
          
            n
          
          
            1
          
        
      
    
    {\displaystyle {\boldsymbol {\Pi }}_{n}^{1}}
  
 subset 
  
    
      
        C
        ⊆
        X
        ×
        Y
      
    
    {\displaystyle C\subseteq X\times Y}
  
 such that 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is the <a href="/facts/Projection_(mathematics)/QxxC9XTp">projection</a> of 
  
    
      
        C
      
    
    {\displaystyle C}
  
 onto 
  
    
      
        X
      
    
    {\displaystyle X}
  
; that is, 
  
    
      
        A
        =
        {
        x
        ∈
        X
        ∣
        ∃
        y
        ∈
        Y
        :
        (
        x
        ,
        y
        )
        ∈
        C
        }
        .
      
    
    {\displaystyle A=\{x\in X\mid \exists y\in Y:(x,y)\in C\}.}
  
</li></ul>
<p>The choice of the Polish space 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 in the third clause above is not very important; it could be replaced in the definition by a fixed <a href="/facts/Uncountable/zl2WqyJQ">uncountable</a> Polish space, say <a href="/facts/Baire_space_(set_theory)/WPtHVtTP">Baire space</a> or <a href="/facts/Cantor_space/h1sNqjBT">Cantor space</a> or the <a href="/facts/Real_line/6eq42xX5">real line</a>.
</p>

Projective hierarchy open-in-new

Projective hierarchy