then f has an extension F to X so that X

Abstract. A classical theorem of Kuratowski says that every Baire one function on a G subspace of a Polish ( = separable completely metriz-able) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a ner gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this hierarchy depending on its oscillation index (f). We prove a re-nement of Kuratowski's theorem: if Y is a subspace of a metric space X and f is a real-valued function on Y such tha