Relatively functionally countable subsets of products

Abstract

A subset $A$ of a topological space $X$ is called relatively functionally countable (RFC) in $X$, if for each continuous function $f : X \to \mathbb{R}$ the set $f[A]$ is countable. We prove that all RFC subsets of a product $\prod\limits_{n\in\omega}X_n$ are countable, assuming that spaces $X_n$ are Tychonoff and all RFC subsets of every $X_n$ are countable. In particular, in a metrizable space every RFC subset is countable. The main tool in the proof is the following result: for every Tychonoff space $X$ and any countable set $Q \subseteq X$ there is a continuous function $f : X^\omega \to \mathbb{R}^2$ such that the restriction of $f$ to $Q^\omega$ is injective.