Extension of fragmented Baire-one functions on Lindelöf spaces (Q1755434)

The main result says that a real-valued Baire-1 function \(f\) on a Lindelöf \(T_{3,5}\)-space \(X\) has Baire-1 extensions to every \(T_{3,5}\)-space \(Y\supset X\), if and only if, \(f\) is fragmented. The notion of functionally countably fragmented functions, which is in general stronger than fragmentability, was introduced in [the authors, ``Extending Baire-one functions on compact spaces'', Preprint, \url{arXiv:1701.00075}], where the authors proved its equivalence with the extendability to \(\beta X\) for Baire-one functions on \(T_{3,5}\)-spaces \(X\). Functional countable fragmentability is an essential tool for the construction of extensions also here. Moreover, a Baire-1 function \(f\) on the uncountable sum \(X\) of compact intervals is constructed such that \(f\) is not functionally countably fragmented. Therefore it cannot be extended to a Baire-1 function on \(\beta X\).