  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.
