Karl Jones: Created page with "In mathematical analysis, a metric space M is called '''complete''' (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, a..."

2016-09-21T21:37:01Z

Created page with "In mathematical analysis, a metric space M is called '''complete''' (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, a..."

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In [[mathematical analysis]], a [[metric space]] M is called '''complete''' (or a Cauchy space) 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.

== Description ==

Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of [[Rational number|rational numbers]] is not complete, because e.g. √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.

== See also ==

* [[Completion (ring theory)]]
* [[Knaster–Tarski theorem]]

== External links ==

* [https://en.wikipedia.org/wiki/Complete_metric_space Complete metric space] @ Wikipedia

Complete metric space - Revision history

Karl Jones: Created page with "In mathematical analysis, a metric space M is called '''complete''' (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, a..."