Domain representability of $C_{\rm p}(X)$
Volume 200 / 2008
Fundamenta Mathematicae 200 (2008), 185-199
MSC: Primary 54C35; Secondary 54E52, 06B35, 05F30.
DOI: 10.4064/fm200-2-5
Abstract
Let $C_{\rm p}(X)$ be the space of continuous real-valued functions on $X$, with the topology of pointwise convergence. We consider the following three properties of a space $X$: (a) $C_{\rm p}(X)$ is Scott-domain representable; (b) $C_{\rm p}(X)$ is domain representable; (c) $X$ is discrete. We show that those three properties are mutually equivalent in any normal $T_1$-space, and that properties (a) and (c) are equivalent in any completely regular pseudo-normal space. For normal spaces, this generalizes the recent result of Tkachuk that $C_{\rm p}(X)$ is subcompact if and only if $X$ is discrete.
