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 ACA0 of
second-order arithmetic, but we find that many unexpected pathologies can occur
in weaker subsystems. In particular, we show that RCA0 does not
prove that every compact discrete countable second-countable space is finite
and that RCA0 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
ACA0