Domain representability of $C_{\rm p}(X)$

Harold Bennett ¹   ; David Lutzer ²

¹ Mathematics Department Texas Tech University Lubbock, TX 79409, U.S.A.
² Mathematics Department College of William and Mary Williamsburg, VA 23187-8795, U.S.A.

Fundamenta Mathematicae, Tome 200 (2008) no. 2, pp. 185-199

Résumé

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.

Détail

DOI : 10.4064/fm200-2-5

Keywords: space continuous real valued functions topology pointwise convergence consider following three properties space nbsp scott domain representable domain representable discrete those three properties mutually equivalent normal space properties equivalent completely regular pseudo normal space normal spaces generalizes recent result tkachuk subcompact only discrete