On extension of continuous mappings to Baire-one mappings
by
Olena Karlova
Chernivtsi National University

Recall that a mapping f:X→ Y between topological spaces is called a Baire-one mapping if it is a pointwise limit of sequence of continuous mappings.

K. Kuratowski [1] proved that if X is a metric space and E is a G_d-subset of X then every Baire-one function f:E→R can be extended to a Baire-one function defined on the whole space.

O. Kalenda and J. Spurný [2] proved the following result: let E be a Lindelöf subspace of a completely regular space X and either E be hereditarily Baire, or E be G_d in X, then every Baire-one function f:E→R can be extended to a Baire-one function on the whole space.

We investigate the possibility of extension of continuous function to a Baire-one function with values in an arbitrary topological space.

A subset E of topological space X is called B₁-retract of X if for an arbitrary space Y every continuous mapping f:E→Y can be extended to a Baire-one mapping g:X→ Y.

Theorem.   Let X be a completely metrizable space, E ⊆ X be an arcwise connected and locally arcwise connected G_d-set. Then E is B₁-retract of X.

1. Kuratowski K., Sur les théorèmes topologiques de la théorie des fonctions de varibles réelles, C.R. Paris, 197, 1933.

2. Kalenda O., Spurný J., Extending Baire-one functions on topological spaces, Topol. Appl. 149 (2005), 195 - 216.

Date received: June 25, 2008