A constructive function-theoretic approach to topological compactness

We introduce 2-compactness, a constructive function-theoretic alternative to topological compactness, based on the notions of Bishop space and Bishop morphism, which are constructive function-theoretic alternatives to topological space and continuous function, respectively. We show that the notion of Bishop morphism is reduced to uniform continuity in important cases, overcoming one of the obstacles in developing constructive general topology posed by Bishop. We prove that 2-compactness generalizes metric compactness, namely that the uniformly continuous real-valued functions on a compact metric space form a 2-compact Bishop topology. Among other properties of 2-compact Bishop spaces, the countable Tychonoff compactness theorem is proved for them. We work within BISH*, Bishop's informal system of constructive mathematics BISH equipped with inductive definitions with rules of countably many premises, a system strongly connected to Martin-Löf's Type Theory.