We show that the first order theory of the lattice of open sets in some natural topological spaces is $m$-equivalent to second order arithmetic. We also show that for many natural computable metric spaces and computable domains the first order theory of the lattice of effectively open sets is undecidable.
Moreover, for several important spaces (e.g., $\mathbb{R}^n$, $n\geq1$, and the domain $P\omega$) this theory is $m$-equivalent to first order arithmetic.

• 03D30 - Other degrees and reducibilities in computability and recursion theory
• 03D45 - Theory of numerations, effectively presented structures
• 03F35 - Second- and higher-order arithmetic and fragments