to [M] was also proven by M. O. Botelho and M. H. Teixeira to imply axioms [K2]–[K4]: Analogous structures Interior, exterior and boundary operators A dual notion to Kuratowski closure operators is that of Kuratowski interior operator, which is a map satisfying the following similar requirements: For these operators, one can reach conclusions that are completely analogous to what was inferred for Kuratowski closures. For example, all Kuratowski interior operators are isotonic, i.e. they satisfy [K4'], and because of intensivity [I2], it is possible to weaken the equality in [I3] to a simple inclusion. The duality between Kuratowski closures and interiors is provided by the natural complement operator on , the map sending . This map is an orthocomplementation on the power set lattice, meaning it satisfies De Morgan's laws: if is an arbitrary set of indices and , By employing these laws, together with the defining properties of , one can show that any Kuratowski interior induces a Kuratowski closure (and vice versa), via the defining relation (and ). Every result obtained concerning may be converted into a result concerning by employing these relations in conjunction with the properties of the orthocomplementation . further provides analogous axioms for Kuratowski exterior operators and Kuratowski boundary operators, which also induce Kuratowski closures via the relations and . Abstract operators Notice that axioms [K1]–[K4] may be adapted to define an abstract unary operation on a general bounded lattice , by formally substituting set-theoretic inclusion with the partial order associated to the lattice, set-theoretic union with the join operation, and set-theoretic intersections with the meet operation; similarly for axioms [I1]–[I4]. If the lattice is orthocomplemented, these two abstract operations induce one another in the usual way. Abstract closure or interior operators can be used to define a generalized topology on the lattice. Since neither unions nor the empty set appear in the requirement for a Moore closure operator, the definition may be adapted to define an abstract unary operator on an arbitrary poset . Connection to other axiomatizations of topology Induction of topology from closure A closure operator naturally induces a topology as follows. Let be an arbitrary set. We shall say that a subset is closed with respect to a Kuratowski closure operator if and only if it is a fixed point of said operator, or in other words it is stable under , i.e. . The claim is that the family