1. Let S(x,8) be a ball with came x and radius 8 in a metric space (X, d). Prove that if 0

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David · Accepted Answer

Theorem 1.5.3: Every metric space has a completion and any two completions are isometric to each 
other. 
1. Prove that any two completions of a metric space (X, d) are isometric to each other. 
Sol:  
Here it is enough to prove that 
If *   (  
    
 )+     *   (  
    
 )+ are two completions of (X, d), then there is a unique 
isometry f from   
         
  such that       . 
 Since   is an isometry,   is one – one. Thus   
     , -    is an isometry from   
  , -       
 Since   is an isometry from X onto   , -    
 . It follows that      
     , -    , -is a 
surjective isometry.  
Let        
  . Then, 
     (     
  )       (  
     )           
Therefore there exists a unique isometry          
       
  which is an extension of g. 
    ,  
    ( )   (  ( ))   (  ( ))      ( )    ( )  
Thus        . There exists a unique isometry h from   
         
  such that         
Hence,

1. Let S(x,8) be a ball with came x and radius 8 in a metric space (X, d). Prove that if 0

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