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 $\omega$-valued functions on $S$ need extend. This last result uses a version of the Isbell-Mrowka space $\Psi$ having a ${C^ * }$-embedded infinite discrete subset. On the other hand, assuming the Product Measure Extension Axiom, no such $\Psi$ exists.