Kuratowski's problem in constructive Topology

A classical result by Kuratowski states that there are at most seven different combinations of the operators of interior and closure in a topological space, which become fourteen if one consider also complement. Two (and hence, usually, more) of these operators can coincide in some special classes of spaces; for instance, Boolean spaces have only six different combinations. This is the classical picture. What happens to this picture if it is looked at from a constructive point of view? The present paper provides an answer to this question, while leaving some problems open. The first part of the paper provides a constructive account of the closure–interior problem and discusses some special classes of spaces. The role of the set-theoretic (pseudo)complement is considered in the second part. The paper ends by showing what the Kuratowski’s problem looks like in a pointfree framework, that is, within the theory of locales. This last part of the paper is independent from the underlying metatheory, although it is obtained by applying the constructive results in the previous parts.