Does adding full induction affect the consistency strength of subsystems of second order arithmetic?

"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 $mathsf{ACA}_0$."

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