Compactness in Constructive Mathematics via Affine Logic

We study topology, particularly compactness, as an extension of Shulman's work on constructive mathematics via affine logic, while allowing propositional impredicativity. We introduce a notion of compactness in affine logic and prove the fundamental properties of compactness, including the extreme value theorem and the Heine-Borel theorem for 'cuts', which are a version of Dedekind cuts in affine logic. Moreover, from the antithesis translation of the Heine-Borel theorem for cuts to intuitionistic logic, we derive the Heine-Borel theorem for one-sided reals intuitionistically, and have verified the proof with an interactive theorem prover. The code is available at https://github.com/hziwara/CutsHeineBorel.