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A note on condensations of function spaces onto $ \sigma$-compact and analytic spaces


Author: Mikołaj Krupski
Journal: Proc. Amer. Math. Soc. 143 (2015), 2263-2268
MSC (2010): Primary 54C35, 54A10, 54D60
DOI: https://doi.org/10.1090/S0002-9939-2015-12507-5
Published electronically: January 16, 2015
MathSciNet review: 3314133
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Abstract: Modifying a construction of W. Marciszewski we prove (in ZFC) that there exists a subspace $ X$ of the real line $ \mathbb{R}$, such that the realcompact space $ C_p(X)$ of continuous real-valued functions on $ X$ with the pointwise convergence topology does not admit a continuous bijection onto a $ \sigma $-compact space. This answers a question of Arhangel'skii.


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Additional Information

Mikołaj Krupski
Affiliation: Institute of Mathematics, Polish Academy of Sciences, Ul. Śniadeckich 8, 00–956 Warszawa, Poland
Email: krupski@impan.pl

DOI: https://doi.org/10.1090/S0002-9939-2015-12507-5
Keywords: Function space, pointwise convergence topology, $C_p(X)$ space, condensation, realcompact space
Received by editor(s): November 15, 2013
Published electronically: January 16, 2015
Additional Notes: The author was partially supported by the Polish National Science Center research grant UMO-2012/07/N/ST1/03525
Communicated by: Mirna Dzamonja
Article copyright: © Copyright 2015 American Mathematical Society