Factoring Out Intuitionistic Theorems: Continuity Principles and the Uniform Continuity Theorem

Abstract

We prove the equivalence between some intuitionistic theorems and the conjunction of a continuity principle and a compactness principle over Bishop's Constructive Mathematics within the programme of Constructive Reverse Mathematics. To clarify our line of thought, we first point out the relation between quasi-equicontinuity, quasi-uniform convergence, and the continuity principle saying that the limit of a convergent sequence of continuous functions is again continuous. Finally, as a spin-off, we conclude that we have found a new, more economic proof of the statement that every convergent sequence of functions on a compact metric space converges uniformly.