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• # Artykuł - szczegóły

## Fundamenta Mathematicae

2006 | 192 | 2 | 183-193

## Extension of functions with small oscillation

EN

### Abstrakty

EN
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 < ω₁, 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.

183-193

wydano
2006

### Twórcy

autor
• Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
autor
• Mathematics and Mathematics Education, National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616