A note on Automatic Baire property

TL;DR

This paper investigates the Automatic Baire property in the context of subsets of the Cantor space accepted by finite automata, revealing limitations and the impact of increased computational power.

Contribution

It characterizes the class of sets with the Automatic Baire property and shows how slight computational enhancements extend beyond this class.

Findings

Not all finite subsets have the Automatic Baire property.

Increasing computational power can produce sets outside the class .

The class  is sensitive to the computational capabilities of automata.

Abstract

Automatic Baire property is a variant of the usual Baire property which is fulfilled for subsets of the Cantor space accepted by finite automata. We consider the family $\mathcal{A}$ of subsets of the Cantor space having the Automatic Baire property. In particular we show that not all finite subsets have the Automatic Baire property, and that already a slight increase of the computational power of the accepting device may lead beyond the class $\mathcal{A}$.