of all subsets of the total space that are complements of closed sets satisfies the three usual requirements for a topology, or equivalently, the family of all closed sets satisfies the following: Notice that, by idempotency [K3], one may succinctly write . [T1] By extensivity [K2], and since closure maps the power set of into itself (that is, the image of any subset is a subset of ), we have . Thus . The preservation of the empty set [K1] readily implies . [T2] Next, let be an arbitrary set of indices and let be closed for every . By extensivity [K2], . Also, by isotonicity [K4'], if for all indices , then for all , which implies . Therefore, , meaning . [T3] Finally, let be a finite set of indices and let be closed for every . From the preservation of binary unions [K4], and using induction on the number of subsets of which we take the union, we have . Thus, . Induction of closure from topology Conversely, given a family satisfying axioms [T1]–[T3], it is possible to construct a Kuratowski closure operator in the following way: if and is the inclusion upset of , then defines a Kuratowski closure operator on . [K1] Since , reduces to the intersection of all sets in the family ; but by axiom [T1], so the intersection collapses to the null set and [K1] follows. [K2] By definition of , we have that for all , and thus must be contained in the intersection of all such sets. Hence follows extensivity [K2]. [K3] Notice that, for all , the family contains itself as a minimal element w.r.t. inclusion. Hence , which is idempotence [K3]. [K4'] Let : then , and thus . Since the latter family may contain more elements than the former, we find , which is isotonicity [K4']. Notice that isotonicity implies and , which together imply . [K4] Finally, fix . Axiom [T2] implies ; furthermore, axiom [T2] implies that . By extensivity [K2] one has and , so that . But , so that all in all . Since then is a minimal element of w.r.t. inclusion, we find . Point 4. ensures additivity [K4]. Exact correspondence between the two structures In fact, these two complementary constructions are inverse to one another: if is the collection of all Kuratowski closure operators on , and is the