First Order Theories of Some Lattices of Open Sets
Oleg V. Kudinov, Victor Selivanov · Logical Methods in Computer Science · 2017
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.