In this paper, we study the relation between w-limit sets and minimal sets. It is known that every minimal set is an w-limit set. However, not every w-limit set is a minimal set, although it contains a minimal set. We would like to know under what condition, an w-limit set turns out to be a minimal set. We prove a necessary and sufficient condition under which every w-limit set is minimal. We also establish a sufficient condition for an w-limit set to be minimal.

periodic points, recurrent points, w-limit set, minimal set.