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 $f:X\to R$ the set $f\left[A\right]$ is countable. We prove that all RFC subsets of a product $\prod _{n\in \omega }{X}_n$ 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 $f:X^{\omega }\to R^2$ such that the restriction of f to $Q^{\omega }$ is injective.