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# Amsterdam Properties of $C_p(X)$ Imply Discreteness of $X$

Published:2008-12-01
Printed: Dec 2008
• D. J. Lutzer
• J. van Mill
• V. V. Tkachuk
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## Abstract

We prove, among other things, that if $C_p(X)$ is subcompact in the sense of de Groot, then the space $X$ is discrete. This generalizes a series of previous results on completeness properties of function spaces.
 Keywords: regular filterbase, subcompact space, function space, discrete space
 MSC Classifications: 54B10 - Product spaces 54C05 - Continuous maps 54D30 - Compactness