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Complete metric space
From Citizendium, the Citizens' Compendium
In mathematics, a complete metric space is a metric space in which every Cauchy sequence in that space is convergent. In other words, every Cauchy sequence in the metric space tends in the limit to a point which is again an element of that space. Hence the metric space is, in a sense, "complete."
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Formal definition
Let X be a metric space with metric d. Then X is complete if for every Cauchy sequence there is an associated element such that .
Examples
- The real numbers R, and more generally finite-dimensional Euclidean spaces, with the usual metric are complete.
- Any compact metric space is sequentially compact and hence complete. The converse does not hold: for example, R is complete but not compact.
- In a space with the discrete metric, the only Cauchy sequences are those which are constant from some point on. Hence any discrete metric space is complete.
- The rational numbers Q are not complete. For example, the sequence (x_{n}) defined by x_{0} = 1, x_{n+1} = 1 + 1/x_{n} is Cauchy, but does not converge in Q.
Completion
Every metric space X has a completion which is a complete metric space in which X is isometrically embedded as a dense subspace. The completion has a universal property.
Examples
- The real numbers R are the completion of the rational numbers Q with respect to the usual metric of absolute distance.
Topologically complete space
Completeness is not a topological property: it is possible for a complete metric space to be homeomorphic to a metric space which is not complete. For example, the map
is a homeomorphism between the complete metric space R and the incomplete space which is the unit circle in the Euclidean plane with the point (0,-1) deleted. The latter space is not complete as the non-Cauchy sequence corresponding to t=n as n runs through the positive integers is mapped to a non-convergent Cauchy sequence on the circle.
We can define a topological space to be metrically topologically complete if it is homeomorphic to a complete metric space. A topological condition for this property is that the space be metrizable and an absolute G_{δ}, that is, a G_{δ} in every topological space in which it can be embedded.