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 Q with the usual absolute value distance. Like every metric space, 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 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.