Extending discrete-valued functions

Abstract

In this paper, we show that for a separable metric space X, every continuous function from a subset S of X into a finite discrete space extends to a continuous function on X if and only if every continuous function from S into any discrete space extends to a continuous function on X. We also show that if there is no inner model having a measurable cardinal, then there is a metric space X with a subspace S such that every 2-valued continuous function from S extends to a continuous function on all of X, but not every discrete-valued continuous function on S extends to such a map on X. Furthermore, if Martin's Axiom is assumed, such a space can be constructed so that not even co-valued functions on S need extend. This last result uses a version of the Isbell-Mrowka space 'P having a C· -embedded infinite discrete subset. On the other hand, assuming the Product Measure Extension Axiom, no such 'P exists.