Application of Hurewicz theorem to classification of Π₁¹ -complete sets

We present a simple proof of a dichotomy theorem by Kechris- -Louveau-Woodin [Kechris, A. S., Louveau, A.: Descriptive set theory and thestructure of sets of uniqueness, London Math. Soc. Lecture Note Ser. 128 (1987), 104-138] telling that every Π₁¹σ-ideal of compact subsets of a Polish space is either Gδ or else Π₁¹ -complete.