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

Volume 200 / 2008

Harold Bennett, David Lutzer Fundamenta Mathematicae 200 (2008), 185-199 MSC: Primary 54C35; Secondary 54E52, 06B35, 05F30. DOI: 10.4064/fm200-2-5


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.


  • Harold BennettMathematics Department
    Texas Tech University
    Lubbock, TX 79409, U.S.A.
  • David LutzerMathematics Department
    College of William and Mary
    Williamsburg, VA 23187-8795, U.S.A.

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