Metrizable space

<p>In <a href="/facts/Topology/2EqW47cU">topology</a> and related areas of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a metrizable space is a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> that is <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphic</a> to a <a href="/facts/Metric_space/gnBBRXtc">metric space</a>. That is, a topological space 
  
    
      
        (
        X
        ,
        τ
        )
      
    
    {\displaystyle (X,\tau )}
  
 is said to be metrizable if there is a <a href="/facts/Metric_(mathematics)/RkUptSo8">metric</a> 
  
    
      
        d
        :
        X
        ×
        X
        →
        [
        0
        ,
        ∞
        )
      
    
    {\displaystyle d:X\times X\to [0,\infty )}
  
 such that the topology induced by 
  
    
      
        d
      
    
    {\displaystyle d}
  
 is 
  
    
      
        τ
        .
      
    
    {\displaystyle \tau .}
  
 <i>Metrization theorems</i> are <a href="/facts/Theorem/f2Pe1FMD">theorems</a> that give <a href="/facts/Sufficient_condition/G45aDCY9">sufficient conditions</a> for a topological space to be metrizable.
</p>

Metrizable space open-in-new

Metrizable space