No abstract

No abstract

Bishop’s presentation of his informal system of constructive mathematics BISH was on purpose closer to the proof-irrelevance of classical mathematics, although a form of proof-relevance was evident in the use of several notions of moduli (of convergence, of uniform continuity, of uniform differentiability, etc.). Focusing on membership and equality conditions for sets given by appropriate existential formulas, we define certain families of proof sets that provide a BHK-interpretation of formulas that correspond to the standard atomic formulas of a first-order theory, within Bishop set theory $(\mathrm{BST})$ , our minimal extension of Bishop’s theory of sets. With the machinery of the general theory of families of sets, this BHK-interpretation within BST is extended to complex formulas. Consequently, we can associate to many formulas $\phi$ of BISH a set ${\texttt{Prf}}(\phi)$ of “proofs” or witnesses of $\phi$ . Abstracting from several examples of totalities in BISH, we define the notion of a set with a proof-relevant equality, and of a Martin-Löf set, a special case of the former, the equality of which corresponds to the identity type of a type in intensional Martin-Löf type theory $(\mathrm{MLTT})$ . Through the concepts and results of BST notions and facts of MLTT and its extensions (either with the axiom of function extensionality or with Vooevodsky’s axiom of univalence) can be translated into BISH. While Bishop’s theory of sets is standardly understood through its translation to MLTT, our development of BST offers a partial translation in the converse direction.

We develop the basic constructive theory of embeddings of Bishop spaces in parallel to the basic classical theory of embeddings of topological spaces. The theory of Bishop spaces is a constructive approach to point-function topology and a natural constructive alternative to the classical theory of the rings of continuous functions. Our most significant result is the translation of the classical Urysohn extension theorem within the theory of Bishop spaces. The related theory of the zero sets of a Bishop topology is also included. We work within $\textrm{BISH}^{\ast }$, Bishop’s informal system of constructive mathematics $\textrm{BISH}$ equipped with inductive definitions with rules of countably many premises.

We introduce cs-topologies, or topologies of 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 simulataneously an open set, the first-component of an open complemented subset, and its 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 mudulus 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 an open subset, given a base for the cs-topology, makes possible to define the notion of uniform continuity of functions between csb-spaces, that is cs-spaces with a given base. In this way, the notions of pointwise and uniform continuity of functions between metric spaces are directly generalised to the notions of pointwise and uniform continuity between csb-topological spaces. Constructions and facts from the classical theory of topologies of open subsets are translated to our constructive theory of topologies of open complemented subsets.