be written as a union of intervals of the form ), this means that is neither open nor closed. If a topological space is endowed with the discrete topology (so that by definition, every subset of is open) then every subset of is a clopen subset. For a more advanced example reminiscent of the discrete topology, suppose that is an ultrafilter on a non-empty set Then the union is a topology on with the property that non-empty proper subset of is an open subset or else a closed subset, but never both; that is, if (where ) then of the following two statements is true: either (1) or else, (2) Said differently, subset is open or closed but the subsets that are both (i.e. that are clopen) are and Regular open sets A subset of a topological space is called a if or equivalently, if where (resp. ) denotes the topological boundary (resp. interior, closure) of in A topological space for which there exists a base consisting of regular open sets is called a . A subset of is a regular open set if and only if its complement in is a regular closed set, where by definition a subset of is called a if or equivalently, if Every regular open set (resp. regular closed set) is an open subset (resp. is a closed subset) although in general, the converses are true. Properties The union of any number of open sets, or infinitely many open sets, is open. The intersection of a finite number of open sets is open. A complement of an open set (relative to the space that the topology is defined on) is called a closed set. A set may be both open and closed (a clopen set). The empty set and the full space are examples of sets that are both open and closed. Uses Open sets have a fundamental importance in topology. The concept is required to define and make sense of topological space and other topological structures that deal with the notions of closeness and convergence for spaces such as metric spaces and uniform spaces. Every subset A of a topological space X contains a (possibly empty) open set; the maximum (ordered under inclusion) such open set is called the interior of A. It can be constructed by taking the union of all the open sets contained in A. A function between two topological