Complete Metric Space

Complete Metric Space

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.

Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because e.g. 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 holes", leading to the completion of a given space, as explained below.

Read more about Complete Metric Space:  Examples, Some Theorems, Completion, Topologically Complete Spaces, Alternatives and Generalizations

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Famous quotes containing the words space and/or complete:

    The secret ones around a stone
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    Allen Tate (1899–1979)

    The history of any nation follows an undulatory course. In the trough of the wave we find more or less complete anarchy; but the crest is not more or less complete Utopia, but only, at best, a tolerably humane, partially free and fairly just society that invariably carries within itself the seeds of its own decadence.
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