Extension of Functions with Small Oscillation

A classical theorem of Kuratowski says that every Baire one function on a Gδ subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this heirarchy depending on its oscillation index β(f). We prove a refinement of Kuratowski’s theorem: if Y is a subspace of a metric space X and f is a real-valued function on Y such that βY (f) < ω , α < ω1, then f has an extension F onto X so that βX (F ) ≤ ω . We also show that if f is a continuous real valued function on Y, then f has an extension F onto X so that βX (F ) ≤ 3. An example is constructed to show that this result is optimal. Let X be a topological space. A real-valued function on X belongs to Baire class one if it is the pointwise limit of a sequence of continuous functions. If X is a Polish (= separable completely metrizable) space, then a classical theorem of Kuratowski [7] states that every Baire one function on a Gδ subspace of X can be extended to a Baire one function on X. In [5], Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes using the oscillation index β, whose definition we now recall. Let X be a topological space and let C denote the collection of all closed subsets of X. A derivation on C is a map D : C → C such that D(P ) ⊆ P for all P ∈ C. The oscillation index β is associated with a family of derivations. Let ε > 0 and a function f : X → R be given. For any P ∈ C, let D1(f, ε, P ) be the set of all x ∈ P such that for any neighborhood U of x, there exist x1, x2 ∈ P ∩U such that |f(x1)− f(x2)| ≥ ε. The derivation D 1(f, ε, ·) may be iterated in the usual manner. For all α < ω1, let D(f, ε, P ) = D(f, ε,D(f, ε, P )). If α is a countable limit ordinal, set D(f, ε, P ) = ∩γ<αD (f, ε, P ). If D(f, ε, P ) 6= ∅ for all α < ω1, let βX(f, ε) = ω1. Otherwise, let βX(f, ε) be the smallest countable ordinal α such that D(f, ε, P ) = ∅. The oscillation index of f is βX(f) = supε>0 βX(f, ε). 1991 Mathematics Subject Classification. Primary 26A21; Secondary 03E15, 54C30.