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Pseudocompact spaces $X$ and $df$-spaces $C_{c}\left( X\right) $

Authors: Jerzy Kakol, Stephen A. Saxon and Aaron R. Todd
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Journal: Proc. Amer. Math. Soc. 132 (2004), 1703-1712
MSC (2000): Primary 46A08, 46A30, 54C35
Published electronically: January 9, 2004
MathSciNet review: 2051131
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Abstract: Let $X$ be a completely regular Hausdorff space, and let $C_{c}\left( X\right) $be the space $C\left( X\right) $ of continuous real-valued functions on $X$endowed with the compact-open topology. We find various equivalent conditions for $C_{c}\left( X\right) $ to be a $df$-space, resolving an old question of Jarchow and consolidating work by Jarchow, Mazon, McCoy and Todd. Included are analytic characterizations of pseudocompactness and an example that shows that, for $C_{c}\left( X\right) $, Grothendieck's $DF$-spaces do not coincide with Jarchow's $df$-spaces. Any such example necessarily answers a thirty-year-old question on weak barrelledness properties for $C_{c}\left( X\right) $, our original motivation.

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Additional Information

Jerzy Kakol
Affiliation: Department of Mathematics, Baruch College, C.U.N.Y., New York, New York 10010, and Faculty of Mathematics and Informatics, A. Mickiewicz University, 60-769 Poznań, Matejki 48-49, Poland

Stephen A. Saxon
Affiliation: Department of Mathematics, University of Florida, P.O. Box 118105, Gainesville, Florida 32611-8105

Aaron R. Todd
Affiliation: Department of Mathematics, Baruch College, C.U.N.Y., New York, New York 10010

Keywords: Compact-open topology, $DF\,$- and $df$-spaces, completely regular, docile, locally complete, weak barrelledness
Received by editor(s): December 12, 2002
Published electronically: January 9, 2004
Additional Notes: We thank Baruch College of New York for their hospitality and support, particularly in the form of a Weissman Visiting Professorship awarded to the first author, also supported by Komitet Banań Naukowych (State Committee for Scientific Research), Poland, grant no. 2803A 022 25.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2004 American Mathematical Society

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