Topologies of open complemented subsets

Abstract

We introduce cs-topologies, or topologies of open complemented subsets, as a new approach to constructive topology that preserves the duality between open and closed subsets of classical topology. Complemented subsets were used successfully by Bishop in his constructive formulation of the Daniell approach to measure and integration. Here we use complemented subsets in topology, in order to describe simultaneously an open set, the first-component of an open complemented subset, together with its given complement as a closed set, the second component of an open complemented subset. We analyse the canonical cs-topology induced by a metric, and we introduce the notion of a modulus of openness for a cs-open subset of a metric space. Pointwise and uniform continuity of functions between metric spaces are formulated with respect to the way these functions inverse open complemented subsets together with their moduli of openness. The addition of moduli of openness in the concept of a complemented open subset, given a base for the cs-topology, makes possible to define the notions of pointwise-like and uniform-like continuity of functions between csb-spaces, that is cs-spaces with a given base.