Computability of Separation Axioms in Countable Second Countable Spaces

Abstract

We analyze the effective content of countable, second countable topological spaces by directly calculating the complexity of several topologically defined index sets. We focus on the separation principles, calibrating an arithmetic completeness result for each of the Tychonoff separation axioms. Beyond this, we prove completeness results for various other topological properties, such as being Polish and having a particular Cantor-Bendixson rank, using tools from computable structure theory. This work contrasts with previous work analyzing countable, second countable spaces which used the framework of reverse mathematics, as reverse mathematics generally lacks the precision to pin down exact arithmetic complexity levels for properties of interest.