# Definitions Let $(X,d)$ be a [[Metrics|metric space]]. Then $(X,d)$ is **complete** iff every [[Cauchy sequences|Cauchy sequence]] is [[Convergence|convergent]], i.e. it has a limit in $X.$ Note that the open interval $(0,1)$, which is not complete, is homeomorphic to $\mathbb{R}$, which is complete, showing that **completeness is not a topological property**. ###### Complete metric subspaces See [[Subspace of a complete metric space is complete if and only if it is closed]]. # Examples We know that $\mathbb{R}$ is complete (see [[Completeness of real numbers]]) Also $\mathbb{R}^{n}$ is complete. See [[Product of complete spaces is complete]]. $\ell^{p}$ is complete. See [[The Space of Continuous & Bounded real-valued Functions on non-empty set is Complete wrt Supremum Norm]]. In general can construct a complete metric space from any metric space as quotient space of Cauchy sequences. (TBC: compare analysis 1 definition of reals versus cantor's construction leading to a discussion of universal property of the completion). # Applications See [[Contraction mapping theorem]]. See Baire category theorem.