Extension of functions with small oscillation

Abstract

Abstract. A classical theorem of Kuratowski says that every Baire one function on a G δ 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 hierarchy 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 β Y (f ) < ω α , α < ω 1 , then f has an extension F to X so that β X (F ) ≤ ω α . We also show that if f is a continuous real-valued function on Y, then f has an extension F to X so that β X (F ) ≤ 3. An example is constructed to show that this result is optimal.

On Minimum and Maximum of Functions of Small Baire Classes

“…Following [5], we define a derivation G which closely related to D. Given a real-valued function f on X, ε > 0, and a closed subset H of X. Define G(f, ε, H) to be the set of all x ∈ H such that for any open neighborhood U of x, there exists x ∈ H ∩ U such that…”

“…The relationship between derivations D and G is given in the following lemma that can be seen in [5,Lemma 4].…”

Sifat-sifat Fungsi Terfragmentasi dan Fungsi Terfragmentasi Terhitung Fungsional

“…Salah satu teorema yang terkenal adalah Teorema Perluasan Tietze untuk fungsi kontinu. Selain fungsi kontinu, beberapa peniliti juga meneliti perluasan fungsi Baire-1, diantaranya [3], [4], [7], [8], dan [9]. Fungsi f : X → Y dengan X dan Y ruang topologi disebut fungsi Baire-1 jika terdapat barisan fungsi kontinu {f n } n∈N dari X ke Y yang konvergen titik demi titik ke f .…”