set contains its own boundary. In other words, if you are "outside" a closed set, you may move a small amount in any direction and still stay outside the set. Note that this is also true if the boundary is the empty set, e.g. in the metric space of rational numbers, for the set of numbers of which the square is less than Any intersection of any family of closed sets is closed (this includes intersections of infinitely many closed sets) The union of closed sets is closed. The empty set is closed. The whole set is closed. In fact, if given a set and a collection of subsets of such that the elements of have the properties listed above, then there exists a unique topology on such that the closed subsets of are exactly those sets that belong to The intersection property also allows one to define the closure of a set in a space which is defined as the smallest closed subset of that is a superset of Specifically, the closure of can be constructed as the intersection of all of these closed supersets. Sets that can be constructed as the union of countably many closed sets are denoted Fσ sets. These sets need not be closed. Examples The closed interval of real numbers is closed. (See for an explanation of the bracket and parenthesis set notation.) The unit interval is closed in the metric space of real numbers, and the set of rational numbers between and (inclusive) is closed in the space of rational numbers, but is not closed in the real numbers. Some sets are neither open nor closed, for instance the half-open interval in the real numbers. Some sets are both open and closed and are called clopen sets. The ray is closed. The Cantor set is an unusual closed set in the sense that it consists entirely of boundary points and is nowhere dense. Singleton points (and thus finite sets) are closed in T1 spaces and Hausdorff spaces. The set of integers is an infinite and unbounded closed set in the real numbers. If is a function between topological spaces then is continuous if and only if preimages of closed sets in are closed in See also Notes References General topology