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A function space $C_{p}(X)$ without a condensation onto a $\sigma $-compact space

Author: Witold Marciszewski
Journal: Proc. Amer. Math. Soc. 131 (2003), 1965-1969
MSC (2000): Primary 54C35, 54A10
Published electronically: October 18, 2002
MathSciNet review: 1955287
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Abstract: Assuming that the minimal cardinality of a dominating family in $\omega ^{\omega }$ is equal to $2^{\omega }$, we construct a subset $X$ of a real line $\mathbb{R}$ such that the space $C_{p}(X)$ of continuous real-valued functions on $X$ does not admit any continuous bijection onto a $\sigma $-compact space. This gives a consistent answer to a question of Arhangel'skii.

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Additional Information

Witold Marciszewski
Affiliation: Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
Address at time of publication: Faculty of Sciences, Division of Mathematics and Computer Science, Vrije Universiteit, De Boelelaan $1081^{a}$, 1081 HV Amsterdam, The Netherlands

Keywords: Function space, pointwise convergence topology, $C_{p}(X)$, condensation
Received by editor(s): July 2, 2001
Received by editor(s) in revised form: December 4, 2001, and February 8, 2002
Published electronically: October 18, 2002
Additional Notes: The author was supported in part by KBN grant 2 P03A 011 15.
Communicated by: Alan Dow
Article copyright: © Copyright 2002 American Mathematical Society