Kuratowski's problem in constructive Topology
Francesco Ciraulo
Abstract
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 (and constructive) framework, that is, within the (constructive) theory of locales.