Editing Complete metric space (section)

==Examples==

The space <math>\Q</math> of rational numbers, with the standard [[metric (mathematics)|metric]] given by the [[absolute value]] of the [[subtraction|difference]], is not complete. 
Consider for instance the sequence defined by 
:<math>x_1 = 1\;</math> and <math>\;x_{n+1} = \frac{x_n}{2} + \frac{1}{x_n}.</math> 
This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit: If the sequence did have a limit <math>x,</math> then by solving <math>x = \frac{x}{2} + \frac{1}{x}</math> necessarily <math>x^2 = 2,</math> yet no rational number has this property. 
However, considered as a sequence of [[real number]]s, it does converge to the [[irrational number]] <math>\sqrt{2}</math>.

The [[open interval]] {{open-open|0,1}}, again with the absolute difference metric, is not complete either. 
The sequence defined by <math>x_n = \tfrac{1}{n}</math> is Cauchy, but does not have a limit in the given space. 
However the [[closed interval]] [[unit interval|{{closed-closed|0,1}}]] is complete; for example the given sequence does have a limit in this interval, namely zero.

The space <math>\R</math> of real numbers and the space <math>\C</math> of [[complex number]]s (with the metric given by the absolute difference) are complete, and so is [[Euclidean space]] <math>\R^n</math>, with the [[Euclidean distance|usual distance]] metric. 
In contrast, [[dimension (vector space)|infinite-dimensional]] [[normed vector space]]s may or may not be complete; those that are complete are [[Banach space]]s. 
The space C{{closed-closed|''a'', ''b''}} of [[continuous functions on a compact Hausdorff space|continuous real-valued functions on a closed and bounded interval]] is a Banach space, and so a complete metric space, with respect to the [[supremum norm]]. 
However, the supremum norm does not give a norm on the space C{{open-open|''a'', ''b''}} of continuous functions on {{open-open|''a'', ''b''}}, for it may contain [[bounded function|unbounded functions]]. 
Instead, with the [[topological space|topology]] of [[compact convergence]], C{{open-open|''a'', ''b''}} can be given the structure of a [[Fréchet space]]: a [[locally convex topological vector space]] whose topology can be induced by a complete [[Metric space#Normed vector spaces|translation-invariant]] metric.

The space '''Q'''<sub>''p''</sub> of [[p-adic number|''p''-adic numbers]] is complete for any [[prime number]] <math>p.</math> 
This space completes '''Q''' with the ''p''-adic metric in the same way that '''R''' completes '''Q''' with the usual metric.

If <math>S</math> is an arbitrary set, then the set {{math|''S''<sup>'''N'''</sup>}} of all sequences in <math>S</math> becomes a complete metric space if we define the distance between the sequences <math>\left(x_n\right)</math> and <math>\left(y_n\right)</math> to be <math>\tfrac{1}{N}</math> where <math>N</math> is the smallest index for which <math>x_N</math> is [[Distinct (mathematics)|distinct]] from <math>y_N</math> or <math>0</math> if there is no such index. 
This space is [[homeomorphic]] to the [[product topology|product]] of a [[countable]] number of copies of the [[discrete space]] <math>S.</math>

[[Riemannian manifold]]s which are complete are called [[geodesic manifold]]s; completeness follows from the [[Hopf–Rinow theorem]].