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Complete metric space

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  1. In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M or alternatively if every Cauchy sequence in M converges in M. Intuitively, a space is complete if the

    Source: Wikipedia Unhappy with this fact? more info
    created by user picturefactobot on December 29, 2008
  2. For instance, the set of rational numbers is not complete, because \sqrt2 is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it. (See the examples below.) It is always possible to "fill all the

    Source: Wikipedia Unhappy with this fact? more info
    created by user picturefactobot on December 29, 2008
  3. Note that completeness is a property of the metric and not of the topology, meaning that a complete metric space can be homeomorphic to a non-complete one

    Source: Wikipedia Unhappy with this fact? more info
    created by user picturefactobot on December 29, 2008
  4. The space Q of rational numbers, with the standard metric given by the absolute value, is not complete

    Source: Wikipedia Unhappy with this fact? more info
    created by user picturefactobot on December 29, 2008
  5. Every compact metric space is complete

    Source: Wikipedia Unhappy with this fact? more info
    created by user picturefactobot on December 29, 2008
  6. For any metric space M, one can construct a complete metric space M (which is also denoted as M with a bar over it), which contains M as a dense subspace

    Source: Wikipedia Unhappy with this fact? more info
    created by user picturefactobot on December 29, 2008
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