Limits of sequences of continuous functions depending on finitely many coordinates

TL;DR

This paper investigates the limits of Baire class one and lower semicontinuous functions on subspaces of countable perfectly normal products, showing they can be approximated by sequences of finitely coordinate-dependent continuous functions.

Contribution

It proves that Baire one functions are pointwise limits of finitely coordinate-dependent continuous functions, and characterizes when lower semicontinuous functions can be similarly approximated.

Findings

Baire one functions on certain subspaces are limits of finitely coordinate-dependent continuous functions.

Lower semicontinuous functions with a finitely dependent minorant can be approximated by increasing sequences of such functions.

The paper answers open questions from previous research by V. Bykov.

Abstract

We answer two questions from {\it V.Bykov, On Baire class one functions on a product space, Topol. Appl. {199} (2016) 55--62,} and prove that every Baire one function on a subspace of a countable perfectly normal product is the pointwise limit of a sequence of continuous functions, each depending on finitely many coordinates. It is proved also that a lower semicontinuous function on a subspace of a countable perfectly normal product is the pointwise limit of an increasing sequence of continuous functions, each depending on finitely many coordinates, if and only if the function has a minorant which depends on finitely many coordinates.