Application of Hurewicz theorem to Classification of $\Pi_1^1$-complete sets

Abstract

We present a simple proof of a dichotomy theorem by Kechris-Louveau-Woodin \cite{Ke1} telling that every $\Pi_1^1$ $\sigma$-ideal of compact subsets of a~Polish space is either G$_{\delta}$ or else $\Pi_1^1$-complete.