On extension of continuous mappings to F_s-measurable mappings
by
Olena Karlova
Chernivtsi National University

A mapping f:X→ Y from a topological space X to a topological space Y is called F_s-measurable if for every open set V in Y the preimage g^-1(V) is an F_s-set in X.

K. Kuratowski [1] proved that if X is a metric space, Y is a Polish space then every continuous mapping f:E→ Y, E ⊆ X, can be extended to an F_s-measurable mapping g:X→Y.

The following result was established in [2]: let E be a Lindelöf subspace of a completely regular space X, Y be a Polish space and either E be hereditarily Baire, or E be G_d in X, then every F_s-measurable mapping f:E→ Y can be extended to an F_s-measurable mapping g:X→ Y.

We prove that any continuous mapping f:E→ Y on a completely metrizable subspace E of a perfect paracompact space X can be extended to an F_s-measurable mapping g:X→ Y with values in an arbitrary topological space Y.

References:

[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: May 6, 2008