On some class of Borel measurable maps and absolute Borel topological spaces (Q697612)

Borel measurable maps between arbitrary topological spaces (or even between arbitrary metrizable spaces) do not always have a definite (countable) Borel rank. The author defines and studies the class of \(p\)-Borel maps; a map \(f:(X,\tau)\to Y\) is \(p\)-Borel if \(f\)~becomes continuous upon enlarging \(\tau\) by countably many Borel sets \(\{A_n:n\in\omega\}\). The Borel rank of~\(f\) is then no higher than the supremum of the additive ranks of the~\(A_n\). The class of \(p\)-Borel maps is shown to be very well-behaved under various operations. The final part of the paper presents a characterization of completely regular absolute Borel spaces, which is shown to have many classical results a corollaries.