François G. Dorais

Research in Logic and Foundations of Mathematics

Reverse mathematics of compact countable second-countable spaces

By F. G. Dorais

We study the reverse mathematics of the theory of countable second-countable topological spaces, with a focus on compactness. We show that the general theory of such spaces works as expected in the subsystem of second-order arithmetic, but we find that many unexpected pathologies can occur in weaker subsystems. In particular, we show that does not prove that every compact discrete countable second-countable space is finite and that does not prove that the product of two compact countable second-countable spaces is compact. To circumvent these pathologies, we introduce strengthened forms of compactness, discreteness, and Hausdorffness which are better behaved in subsystems of second-order arithmetic weaker than .