A characterization of rings of density continuous functions.

Abstract

A density continuous function is defined as a continuous function from a Tychonoff space $X$ into the real numbers with the density topology. The collection of density continuous functions on $X$ is denoted by $C(X,\mathbb{R}_d)$. It is shown that $C(X,\mathbb{R}_d)$ is a ring precisely when each density continuous function is locally constant, and in this case $X$ is defined to be a density $P$-space. Examples of density $P$-spaces are given.