Eventually different functions and inaccessible cardinals

Abstract

In this paper, we are considering the Baire property of the eventually different topology as a regularity property for sets of reals and investigate the logical strength of the statements "Every $\Delta_2^1$ set has the Baire property in the eventually different topology" and "Every $\Sigma_2^1$ set has the Baire property in the eventually different topology". The latter statement turns out to be equivalent to "$\omega_1$ is inaccessible by reals".