Strong Versus Uniform Continuity: A Constructive Round

The notion of apartness has recently shown promise as a means of lifting constructive topology from the restrictive context of metric spaces to more general settings. Extending the point-subset apartness axiomatised beforehand, we characterise the constructive meaning of 'two subsets of a given set lie apart from each other'. We propose axioms for such apartness relations and verify them for the apartness relation associated with an abstract uniform space. Moreover, we relate uniform continuity to strong continuity, the natural concept for mappings between sets endowed with an apartness structure, which says that if the images of two subsets lie apart from each other, then so do the original subsets. Proofs are carried out with intuitionistic logic, and most of them without the principle of countable choice.