collection of all families consisting of complements of all sets in a topology, i.e. the collection of all families satisfying [T1]–[T3], then such that is a bijection, whose inverse is given by the assignment . First we prove that , the identity operator on . For a given Kuratowski closure , define ; then if its primed closure is the intersection of all -stable sets that contain . Its non-primed closure satisfies this description: by extensivity [K2] we have , and by idempotence [K3] we have , and thus . Now, let such that : by isotonicity [K4'] we have , and since we conclude that . Hence is the minimal element of w.r.t. inclusion, implying . Now we prove that . If and is the family of all sets that are stable under , the result follows if both and . Let : hence . Since is the intersection of an arbitrary subfamily of , and the latter is complete under arbitrary intersections by [T2], then . Conversely, if , then is the minimal superset of that is contained in . But that is trivially itself, implying . We observe that one may also extend the bijection to the collection of all Čech closure operators, which strictly contains ; this extension is also surjective, which signifies that all Čech closure operators on also induce a topology on . However, this means that is no longer a bijection. Examples As discussed above, given a topological space we may define the closure of any subset to be the set , i.e. the intersection of all closed sets of which contain . The set is the smallest closed set of containing , and the operator is a Kuratowski closure operator. If is any set, the operators such that are Kuratowski closures. The first induces the indiscrete topology , while the second induces the discrete topology . Fix an arbitrary , and let be such that for all . Then defines a Kuratowski closure; the corresponding family of closed sets coincides with , the family of all subsets that contain . When , we once again retrieve the discrete topology (i.e. , as can be seen from the definitions). If is an infinite cardinal number such that , then the operator such thatsatisfies all four Kuratowski axioms. If , this operator induces the cofinite topology on ; if , it induces the cocountable