Topological Complexity of omega-Powers: Extended Abstract

Abstract

The operation of taking the omega-power $V^omega$ of a language $V$ is a fundamental operation over finitary languages leading to omega-languages. Since the set $X^omega$ of infinite words over a finite alphabet $X$ can be equipped with the usual Cantor topology, the question of  the topological complexity of omega-powers of finitary  languages naturally arises and has  been posed by Damian Niwinski (1990), Pierre Simonnet (1992), and Ludwig Staiger (1997). We investigate the topological complexity of  omega-powers. We prove the following very surprising results which show that omega-powers exhibit a great opological complexity: for each non-null countable ordinal  $xi$, there  exist some $Sigma^0_xi$-complete omega-powers, and some $Pi^0_xi$-complete omega-powers. On the other hand, the Wadge hierarchy  is a great refinement of the Borel hierarchy, determined by Bill Wadge. We show that, for each ordinal $xi$ greater than or equal to 3, there are uncountably many Wadge degrees of omega-powers of Borel rank $xi +1$. Using tools of effective descriptive set theory, we prove  some effective versions of the above results.