Metric map

<p>In the <a href="/facts/Mathematics/pxTouaz4">mathematical</a> theory of <a href="/facts/Metric_space/gnBBRXtc">metric spaces</a>, a metric map is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> between metric spaces that does not increase any distance. These maps are the <a href="/facts/Morphism/Kdpuyz3S">morphisms</a> in the <a href="/facts/Category_of_metric_spaces/W2kbarEJ">category of metric spaces</a>, Met. Such functions are always  <a href="/facts/Continuous_function/sKbl02pB">continuous functions</a>.
They are also called <a href="/facts/Lipschitz_continuity/Hw20EPEU">Lipschitz functions</a> with <a href="/facts/Lipschitz_constant/Hw20EPEU">Lipschitz constant</a> 1, nonexpansive maps, nonexpanding maps, weak contractions, or short maps.
</p><p>Specifically, suppose that 
  
    
      
        X
      
    
    {\displaystyle X}
  
 and 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 are metric spaces and 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> from 
  
    
      
        X
      
    
    {\displaystyle X}
  
 to 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
. Thus we have a metric map when, <a href="/facts/For_any/agNmvJKN">for any</a> points 
  
    
      
        x
      
    
    {\displaystyle x}
  
 and 
  
    
      
        y
      
    
    {\displaystyle y}
  
 in 
  
    
      
        X
      
    
    {\displaystyle X}
  
,

d
          
            Y
          
        
        (
        f
        (
        x
        )
        ,
        f
        (
        y
        )
        )
        ≤
        
          d
          
            X
          
        
        (
        x
        ,
        y
        )
        .
        
      
    
    {\displaystyle d_{Y}(f(x),f(y))\leq d_{X}(x,y).\!}

Here 
  
    
      
        
          d
          
            X
          
        
      
    
    {\displaystyle d_{X}}
  
 and 
  
    
      
        
          d
          
            Y
          
        
      
    
    {\displaystyle d_{Y}}
  
 denote the metrics on 
  
    
      
        X
      
    
    {\displaystyle X}
  
 and 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 respectively.
</p>

Metric map open-in-new

Metric map