Not open does not imply closed. Most subsets of the real numbers are neither open nor closed (e.g. N, or [0,1), etc.). On the other hand, the set of all real numbers is both open and closed.
If a set A is not open then there exists a point x in A such that x is not an interior point of A. For instance, 0 is not interior to the interval [0,1), so its not open. Yet the interval is still "half-open", so it can't be closed.
blkqi's answers is the one relevant to your question, but going more generally, into the topological definition, a topology is a set S plus a collection of special subsets T which is closed under finite intersections and arbitrary unions, and such that both S and the null set are in T. Then a closed set is one which has an open set as complement. Thus both S and the null set are "clopen", both closed and open. There again you see that "not open" and "closed" are not synonyms.
Taking the standard topology on the real numbers, this reduces to blkqi's answer.