Complete space

:For Cauchy completion in category theory, see Karoubi envelope.

Related Topics:
Category theory - Karoubi envelope

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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.

Related Topics:
Mathematical analysis - Metric space - Cauchy sequence - Limit

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Intuitively, a space is complete if it "doesn't have any holes", if there aren't any "points missing".

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For instance, the rational numbers are not complete, because √2 is "missing" even though you can construct a Cauchy sequence of rational numbers that converge to it. (See the examples below.) It is always possible to "fill all the holes", leading to the completion of a given space, as will be explained below.

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