A constructive Galois connection between closure and interior

We construct a Galois connection between closure and interior operators on a given set. All arguments are intuitionistically valid. Our construction is an intuitionistic version of the classical correspondence between closure and interior operators via complement. In classical mathematics, the theory of closure operators and that of interior operators can be derived one from another. In fact, A is a closure operator if and only if its companion −A− (where − is complementation) is an interior operator. Since passing to the companion is an involution, one derives that J is an interior operator if and only if −J− is a closure operator. ¿From an intuitionistic point of view, the picture is more complex. In fact, −A− is not in general an interior operator. So the notion of companion has to be defined differently. Our proposal is based on the notion of compatibility between two operators on subsets of a given set. We show intuitionistically that every closure operator A has a greatest compatible interior operator J(A). Since classically J(A) = −A−, we choose J(A) as the companion of A. Dually, the companion of an interior operator J is the greatest closure operator A(J ) which is compatible with J . Classically A(J ) = −J−. We prove thatA and J form a Galois connection between closure and interior operators on given set, that is A ⊆ A(J ) if and only if J ⊆ J(A). Classically, this collapses to the triviality A ⊆ −J− if and only if J ⊆ −A−. In section 1, we start by analysing the notion of compatibility between arbitrary operators on the same set. We specialise to the case of compatibility between a closure and an interior operator in section 2. There we present the constructions of A and J and prove that they form a Galois connection. Following [15], a set equipped with both a closure and an interior operator which are compatible is called a basic topology. In section 3, we introduce two classes of basic topologies: saturated basic topologies, in which the reduction is completely determined by the saturation, and reduced ones, symmetrically. We show that the Galois connection can be seen as the composition of two adjunctions between these two classes and all basic topologies. Dipartimento di Matematica, Università di Padova, Via Trieste, 63 I-35121 Padova, Italy, {ciraulo,sambin}@math.unipd.it.