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Let X be a Polish space and Y be a separable metric space. For a fixed ξ < ω₁, consider a family of Baire-ξ functions. Answering a question of Tomasz Natkaniec, we show that if for a function f: X → Y, the set is finite for every x ∈ X, then f itself is necessarily Baire-ξ. The proof is based on a characterization of sets which can be interesting in its own right.
Tamás Mátrai. "On the closure of Baire classes under transfinite convergences." Fundamenta Mathematicae 183.2 (2004): 157-168. <http://eudml.org/doc/283214>.
@article{TamásMátrai2004, abstract = {Let X be a Polish space and Y be a separable metric space. For a fixed ξ < ω₁, consider a family $f_\{α\}: X → Y~(α<ω₁)$ of Baire-ξ functions. Answering a question of Tomasz Natkaniec, we show that if for a function f: X → Y, the set $\{α < ω₁: f_\{α\}(x) ≠ f(x)\}$ is finite for every x ∈ X, then f itself is necessarily Baire-ξ. The proof is based on a characterization of $Σ⁰_\{η\}$ sets which can be interesting in its own right.}, author = {Tamás Mátrai}, journal = {Fundamenta Mathematicae}, keywords = {transfinite convergence; Baire class; closure}, language = {eng}, number = {2}, pages = {157-168}, title = {On the closure of Baire classes under transfinite convergences}, url = {http://eudml.org/doc/283214}, volume = {183}, year = {2004}, }
TY - JOUR AU - Tamás Mátrai TI - On the closure of Baire classes under transfinite convergences JO - Fundamenta Mathematicae PY - 2004 VL - 183 IS - 2 SP - 157 EP - 168 AB - Let X be a Polish space and Y be a separable metric space. For a fixed ξ < ω₁, consider a family $f_{α}: X → Y~(α<ω₁)$ of Baire-ξ functions. Answering a question of Tomasz Natkaniec, we show that if for a function f: X → Y, the set ${α < ω₁: f_{α}(x) ≠ f(x)}$ is finite for every x ∈ X, then f itself is necessarily Baire-ξ. The proof is based on a characterization of $Σ⁰_{η}$ sets which can be interesting in its own right. LA - eng KW - transfinite convergence; Baire class; closure UR - http://eudml.org/doc/283214 ER -