Complete metric space

A complete metric space is a metric space $(X,d)$ in which every Cauchy sequence converges (as a convergent sequence ) to a point of $X$.

Completeness is central in analysis and topology; for example it interacts strongly with compactness (see compact iff complete and totally bounded ) and with the Baire category theorem (see complete metric spaces are Baire ).

Examples:

• $(\mathbb{R}^n,\|\cdot\|_2)$ is complete.
• $(\mathbb{Q},|\cdot|)$ is not complete.