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Complete metric space

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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."

Contents

Formal definition

Let X be a metric space with metric d. Then X is complete if for every Cauchy sequence x_1,x_2,\ldots \in X there is an associated element x \in X such that \mathop{\lim}_{n \rightarrow \infty} d(x_n,x)=0.

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 (xn) defined by x0 = 1, xn+1 = 1 + 1/xn is Cauchy, but does not converge in Q.

Completion

Every metric space X has a completion \bar X 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

 t \leftrightarrow \left(\frac{2t}{1+t^2},\frac{1-t^2}{1+t^2}\right)

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 Gδ, that is, a Gδ in every topological space in which it can be embedded.

See also

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