Some unpublished Recław theorems and their applications to Baire-star-one functions

Tomasz Natkaniec,Waldemar Sieg

doi:10.3906/mat-1811-103

Abstract

Lunina's 7-tuples $\langle E^1,\ldots, E^7\rangle$ of sets of pointwise convergence, divergence to $\infty$, divergence to $-\infty$, etc. for sequences of Baire-star-one functions are cha\-racterized. Generalization on ideal convergence of such sequences is discussed. Limits and ideal limits of sequences of Baire-star-one functions are considered in the last part of the article.

Highlights

Let f⃗ =n be a sequence of real-valued functions defined on a metric space X
It is not difficult to show that if f⃗ is a sequence of continuous functions the set L(f⃗), of all points x ∈ X such that (fn(x))n converges, is of the type Fσδ(X)
The class of continuous functions will be replaced by B1∗(X) : the class of all real Baire-star-one mappings defined on X

Summary

Introduction

It is not difficult to show that if f⃗ is a sequence of continuous functions the set L(f⃗) , of all points x ∈ X such that (fn(x))n converges, is of the type Fσδ(X) . In 2010 Borzestowski and Recław proved [1] an ideal version of Lunina’s theorem for sequences of continuous functions. 3. Lunina’s 7-tuples for the families of Borel functions Theorem 3.1 ([18]) Let X be a metrizable space.

Results

Conclusion