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 $\mathsf{ ACA}_0$ of second-order arithmetic, but we find that many unexpected pathologies can occur in weaker subsystems. In particular, we show that $\mathsf{ RCA}_0$ does not prove that every compact discrete countable second-countable space is finite and that $\mathsf{ RCA}_0$ 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 $\mathsf{ ACA}_0$.
Categories: math.LO