Complete metric space and complete subset

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 $E$.

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 ).