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".
Jörg BRENDLE. Benedikt LÖWE. "Eventually different functions and inaccessible cardinals." J. Math. Soc. Japan 63 (1) 137 - 151, January, 2011. https://doi.org/10.2969/jmsj/06310137