A new characterization of Baire class 1 functions. (English) Zbl 1181.03048

Motto Ros, Luca

Real Anal. Exch. 34(2008-2009), No. 1, 29-48 (2009).

The author gives a new characterization of Baire class \(1\) functions on ultrametric spaces by proving that they are the pointwise limits of sequences of certain simple Lipschitz functions. To this end, the author also recalls several classical characterizations of Baire class \(1\) functions.
Let \((X,d)\) be an ultrametric space and let \(Y\) be a separable metrizable space. A set \(A \subseteq X\) is called full if there is an \(r >0\) such that for every \(x \in A\) we have \(\{ y \in X : d(x,y) < r \} \subseteq A\). A function \(f : X \to Y\) is called full if it has finite range and for every \(y \in Y\), \(f^{-1}(y) \subseteq X\) is a full set.
The main result of the paper is the following.
Theorem. Let \((X,d)\) be an ultrametric space and let \(Y\) be a separable metrizable space. Then \(f : X \rightarrow Y\) is of Baire class \(1\) if and only if \(f\) is the pointwise limit of a sequence of full functions.
The proof of this result can be carried out under ZF and countable choice over the reals; in particular, it is consistent with the Axiom of Determinacy.

Reviewer: Tamás Mátrai (Highland Park)

MSC:

03E15	Descriptive set theory
54H05	Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)

Keywords:

ultrametric space; Baire class 1 function; Delta function; Lipschitz function; pointwise convergence