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Concept
Complete metric space
Summary
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.
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 further examples below). It is always possible to "fill all the holes", leading to the completion of a given space, as explained below.
Cauchy sequence
A sequence in a metric space is called Cauchy if for every positive real number there is a positive integer such that for all positive integers
Complete space
A metric space is complete if any of the following equivalent conditions are satisfied:
Every Cauchy sequence of points in has a limit that is also in
Every Cauchy sequence in converges in (that is, to some point of ).
Every decreasing sequence of non-empty closed subsets of with diameters tending to 0, has a non-empty intersection: if is closed and non-empty, for every and then there is a point common to all sets
The space Q of rational numbers, with the standard metric given by the absolute value of the difference, is not complete.
Consider for instance the sequence defined by and
This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit: If the sequence did have a limit then by solving necessarily yet no rational number has this property.
However, considered as a sequence of real numbers, it does converge to the irrational number .
The open interval , again with the absolute difference metric, is not complete either.
The sequence defined by is Cauchy, but does not have a limit in the given space.
However the closed interval is complete; for example the given sequence does have a limit in this interval, namely zero.
The space R of real numbers and the space C of complex numbers (with the metric given by the absolute difference) are complete, and so is Euclidean space Rn, with the usual distance metric.
Official source
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