On Continuous Extenders

Abstract

The Tietze extension theorem guarantees that if A is a closed subset of a normal space X, then eyery real-valued continuous function f on A can be extended to a continuous real-valued function f on X. Many topologists and analysts studied situations in which the extension process from f to f could be carried out in a natural way. Their early results concerning real-valued functions may be summarized in the Dugundji extension theorem for metric spaces. Borges generalized that theorem to the class of stratifiable spaces. The only other large class for which some version of the Dugundji extension theorem is known is the class of generalized ordered spaces, that is, the class of spaces that can be embedded in linearly ordered spaces with the usual open-interval topology. This chapter presents different characterization. The best known pathological members of this class are the Sorgenfrey line [S] and the Michael line [M-J]. It is somewhat surprising that in order to prove a Dugundji extension theorem for such spaces, one may consider only bounded functions.