Axiom of countable choice

<p>The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an <a href="/facts/Axiom/mLjP14Mc">axiom</a> of <a href="/facts/Axiomatic_set_theory/1ptfsvUP">set theory</a> that states that every <a href="/facts/Countable/Wj4DmWPv">countable</a> collection of <a href="/facts/Empty_set/ffEk3eA6">non-empty</a> <a href="/facts/Set_(mathematics)/BfucfHMq">sets</a> must have a <a href="/facts/Choice_function/AutN9YcF">choice function</a>. That is, given a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> 
  
    
      
        A
      
    
    {\displaystyle A}
  
 with <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> 
  
    
      
        
          N
        
      
    
    {\displaystyle \mathbb {N} }
  
 (where 
  
    
      
        
          N
        
      
    
    {\displaystyle \mathbb {N} }
  
 denotes the set of <a href="/facts/Natural_number/u65og8JE">natural numbers</a>) such that 
  
    
      
        A
        (
        n
        )
      
    
    {\displaystyle A(n)}
  
 is a non-empty set for every 
  
    
      
        n
        ∈
        
          N
        
      
    
    {\displaystyle n\in \mathbb {N} }
  
, there exists a function 
  
    
      
        f
      
    
    {\displaystyle f}
  
 with domain 
  
    
      
        
          N
        
      
    
    {\displaystyle \mathbb {N} }
  
 such that 
  
    
      
        f
        (
        n
        )
        ∈
        A
        (
        n
        )
      
    
    {\displaystyle f(n)\in A(n)}
  
 for every 
  
    
      
        n
        ∈
        
          N
        
      
    
    {\displaystyle n\in \mathbb {N} }
  
.
</p>

Axiom of countable choice open-in-new

Axiom of countable choice