When $C_p(X)$ is domain representable
Volume 223 / 2013
Fundamenta Mathematicae 223 (2013), 65-81
MSC: Primary 54C30; Secondary 54B10, 54E50.
DOI: 10.4064/fm223-1-5
Abstract
Let $M$ be a metrizable group. Let $G$ be a dense subgroup of $M^X$. We prove that if $G$ is domain representable, then $G = M^X$. The following corollaries answer open questions. If $X$ is completely regular and $C_p(X)$ is domain representable, then $X$ is discrete. If $X$ is zero-dimensional, $ T_2$, and $C_p(X,\mathbb {D})$ is subcompact, then $X$ is discrete.
