In mathematics, a set , where is some topological space, is said to be closed if , the complement of in , is an open set. The empty set and the set X itself are always closed sets. The finite union and arbitrary intersection of closed sets are again closed.
Let X be the open interval (0, 1) with the usual topology induced by the Euclidean distance. Open sets are then of the form
As a more interesting example, consider the function space (with a < b). This space consists of all real-valued continuous functions on the closed interval [a, b] and is endowed with the topology induced by the norm