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
Famous quotes containing the words complete and/or space:
“Fate forces its way to the powerful and violent. With subservient obedience it will assume for years dependency on one individual: Caesar, Alexander, Napoleon, because it loves the elemental human being who grows to resemble it, the intangible element. Sometimes, and these are the most astonishing moments in world history, the thread of fate falls into the hands of a complete nobody but only for a twitching minute.”
—Stefan Zweig (18811942)
“At first thy little being came:
If nothing once, you nothing lose,
For when you die you are the same;
The space between, is but an hour,
The frail duration of a flower.”
—Philip Freneau (1752–1832)