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 $\prod _{n\in \ddot{I}‰}{X}_n$ 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 $f:X^{\ddot{I}}‰\to {\mathbb{R}}^2$ such that the restriction of $f$ to $Q^{\ddot{I}}‰$ is injective.

11 pages, minor changes