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 $F$ of a metric space $X$ is closed if $F$ contains all of its limit points; this can be characterized by saying that if a sequence in $F$ converges to a point $x$ in $X$, then $x$ must be in $F$.  It also makes sense to ask whether a subset of $X$ 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 $\mathbb{Q}$ with the usual absolute value distance.  Like every metric space, $\mathbb{Q}$ 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 $[0,1]$, which will be closed in $\mathbb{Q}$ but not complete.
If $X$ is a complete metric space, then a subset of $X$ is closed if and only if it is complete.
