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2010/2011 C(X) and related ideals
S. Afrooz, M. Namdari
Real Anal. Exchange 36(1): 45-54 (2010/2011).


We have characterized the spaces $X$ for which the smallest $z$-ideal containing $\cinfty$ is prime. It turns out that $\cinfty$ is a $z$-ideal in $C(X)$ if and only if every zero-set contained in an open locally compact $\sigma$-compact set is compact. Some interesting ideals related to $\cinfty$ are introduced and corresponding to the relations between these ideals and $\cinfty$, topological spaces $X$ are characterized. Some compactness concepts are explicitly stated in terms of ideals related to $\cinfty$. Finally we have shown that a $\sigma$-compact space $X$ is Baire \ifif every ideal containing $\cinfty$ is essential.


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S. Afrooz. M. Namdari. "C(X) and related ideals." Real Anal. Exchange 36 (1) 45 - 54, 2010/2011.


Published: 2010/2011
First available in Project Euclid: 14 March 2011

zbMATH: 1245.54019
MathSciNet: MR3016402

Primary: 26A04 , 54C40
Secondary: 26A05

Keywords: $\ck$, $\sigma$-compact , $C__\infty(X)$ , Baire , Locally Compact

Rights: Copyright © 2010 Michigan State University Press

Vol.36 • No. 1 • 2010/2011
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