# Difference between complete and closed set

What is the difference between a complete metric space and a closed set?

Can a set be closed but not complete?

A metric space is complete if every Cauchy sequence converges (to a point already in the space). A subset $$FF$$ of a metric space $$XX$$ is closed if $$FF$$ contains all of its limit points; this can be characterized by saying that if a sequence in $$FF$$ converges to a point $$xx$$ in $$XX$$, then $$xx$$ must be in $$FF$$. It also makes sense to ask whether a subset of $$XX$$ is complete, because every subset of a metric space is a metric space with the restricted metric.
It turns out that a complete subspace must be closed, which essentially results from the fact that convergent sequences are Cauchy sequences. However, closed subspaces need not be complete. For a trivial example, start with any incomplete metric space, like the rational numbers $$Q\mathbb{Q}$$ with the usual absolute value distance. Like every metric space, $$Q\mathbb{Q}$$ is closed in itself, so there you have a subset that is closed but not complete. If taking the whole space seems like cheating, just take the rationals in $$[0,1][0,1]$$, which will be closed in $$Q\mathbb{Q}$$ but not complete.
If $$XX$$ is a complete metric space, then a subset of $$XX$$ is closed if and only if it is complete.