Complete metric space

Complete metric space

In <a href="/en/Mathematical_analysis" title="Mathematical analysis" class="wl">mathematical analysis</a>, a <a href="/en/Metric_space" title="Metric space" class="wl">metric space</a> M is called complete (or a Cauchy space) if every <a href="/en/Cauchy_sequence#In_a_metric_space" title="Cauchy sequence" class="wl">Cauchy sequence</a> of points in M has a <a href="/en/Limit_of_a_sequence" title="Limit of a sequence" class="wl">limit</a> that is also in M.

<div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none" about="#mwt1">Metric geometry</div>
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Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of <a href="/en/Rational_number" title="Rational number" class="wl">rational numbers</a> is not complete, because e.g. <a href="/en/Square_root_of_2" title="Square root of 2" class="wl"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}">
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 <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation>
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</math><img loading="lazy"src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\sqrt {2}}"></a> is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it (see further examples below). It is always possible to "fill all the holes", leading to the completion of a given space, as explained below.

In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if there are no "points missing" from it . For instance, the set of rational numbers is not complete, because e.g. 
  
    
      
        
          
            2
          
        
      
    
    {\displaystyle {\sqrt {2}}}
  
 is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it . It is always possible to "fill all the holes", leading to the completion of a given space, as explained below. 