Topological extension of parity automata

The paper presents a concept of a coloring - an extension of deterministic parity automata. A coloring K is a function A^@?->N [email protected]?"@a"@?"A"^"@wliminfn->~K(@[email protected]?"n) ~K(@[email protected]?"n)=0mod2}. We show that sets defined by colorings are exactly all @D"3^0 sets in the standard product topology on A^@w. Furthermore, when considering natural subfamilies of all colorings, we obtain families BC(@S"2^0), @D"2^0, and BC(@S"1^0). Therefore, colorings can be seen as a characterisation of all these classes with a uniform acceptance condition. Additionally, we analyse a similar definition of colorings where the limsup condition is used instead of liminf. It turns out that such colorings have smaller expressive power - they cannot define all @D"3^0 sets.