Open sets appear directly in the definition of a topological space. It next seems that closed sets are needed. Suppose is a topological space. A subset is defined to be a closed set if and only if is an open set. Thus, the complement of any open set is closed, and the complement of any closed set is open. Any closed interval, such as , is a closed set because its complement, , is an open set. For another example, is an open set; therefore, is a closed set. The use of ``'' may seem wrong in the last expression, but ``'' cannot be used because and do not belong to . Thus, the use of ``'' is just a notational quirk.
Are all subsets of either closed or open? Although it appears that open sets and closed sets are opposites in some sense, the answer is no. For , the interval is neither open nor closed (consider its complement: is closed, and is open). Note that for any topological space, and are both open and closed!
Steven M LaValle 2012-04-20