| Abstract
| - The third author noticed in his 1992 Ph.D. thesis [P. Simonnet, Automates et théorie descriptive (1992).] that every regular tree language of infinite trees is in a class \hbox{$\Game (D_n({\bf \Sigma}_2^0))$} for some natural number n ≥ 1 , where \hbox{$\Game}$} is the game quantifier. We first give a detailed exposition of this result. Next, using an embedding of the Wadge hierarchy of non self-dual Borel subsets of the Cantor space 2 ω into the class ∆21, and the notions of Wadge degree and Veblen function, we argue that this upper bound on the topological complexity of regular tree languages is much better than the usual ∆21.
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