Complete metric space and complete subset - Imported Knowlpedia Content

A metric space $(X,d)$ is complete if every Cauchy sequence $(x_n)$ in $X$ converges to some point $x\in X$ .

A subset $E\subset X$ is called complete if the restricted metric space $(E,d|_{E\times E})$ is complete; equivalently, every Cauchy sequence in $E$ converges to a point of $E$ .

Remarks

Context. Completeness is the property that "no limit points are missing." It is central in analysis (e.g., for existence theorems based on Cauchy sequences).

Examples

$(\mathbb{R}^k,d)$ with the usual Euclidean distance is complete (see Completeness of R^k).
The open interval $(0,1)\subset\mathbb{R}$ with the usual distance is not complete: the Cauchy sequence $x_n=1/n$ converges to $0\notin(0,1)$ .
If $X$ is complete and $E\subset X$ is closed, then $E$ is complete (see closed subsets of complete spaces are complete).