Complete metric spaces

Definition. A metric space is complete if every Cauchy sequence has a limit in the space.

Complete. $\mathbb{R}$, $\mathbb{R}^n$ with Euclidean metric, closed subsets of $\mathbb{R}^n$.

Not complete. $\mathbb{Q}$ (with usual metric: $1,1.4,1.41,\ldots$ converges to $\sqrt2\notin\mathbb{Q}$); $]0,1]$ (sequence $1/n$ is Cauchy but converges to $0\notin]0,1]$).

Conclusion. Completeness depends both on the metric and the underlying set.