## Oleg Kudinov ; Victor Selivanov - First Order Theories of Some Lattices of Open Sets

lmcs:3659 - Logical Methods in Computer Science, August 25, 2017, Volume 13, Issue 3 - https://doi.org/10.23638/LMCS-13(3:16)2017
First Order Theories of Some Lattices of Open Sets

Authors: Oleg Kudinov ; Victor Selivanov

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.

Volume: Volume 13, Issue 3
Published on: August 25, 2017
Accepted on: August 25, 2017
Submitted on: August 25, 2017
Keywords: Mathematics - Logic,Computer Science - Logic in Computer Science,03D78, 03D45, 03D55, 03D30