Embeddings of Bishop spaces

Abstract

Abstract 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.