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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Covering an uncountable square by countably many continuous functions


Authors: Wiesław Kubiś and Benjamin Vejnar
Journal: Proc. Amer. Math. Soc. 140 (2012), 4359-4368
MSC (2010): Primary 03E05, 03E15; Secondary 54H05
Published electronically: May 1, 2012
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Abstract: We prove that there exists a countable family of continuous real functions whose graphs, together with their inverses, cover an uncountable square, i.e. a set of the form $ X\times X$, where $ X\subseteq \mathbb{R}$ is uncountable. This extends Sierpiński's theorem from 1919, saying that $ S\times S$ can be covered by countably many graphs of functions and inverses of functions if and only if $ \vert S\vert\leqslant \aleph _1$. Using forcing and absoluteness arguments, we also prove the existence of countably many $ 1$-Lipschitz functions on the Cantor set endowed with the standard non-archimedean metric that cover an uncountable square.


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Additional Information

Wiesław Kubiś
Affiliation: Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic – and – Institute of Mathematics, Jan Kochanowski University in Kielce, Poland
Email: kubis@math.cas.cz

Benjamin Vejnar
Affiliation: Department of Mathematical Analysis, Charles University, Prague, Czech Republic

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11292-4
PII: S 0002-9939(2012)11292-4
Keywords: Uncountable square, covering by continuous functions, set of cardinality $ℵ_{1}$
Received by editor(s): January 11, 2010
Received by editor(s) in revised form: June 6, 2011
Published electronically: May 1, 2012
Additional Notes: The research of the first author was supported in part by Grant IAA 100 190 901 and by the Institutional Research Plan of the Academy of Sciences of Czech Republic, No. AVOZ 101 905 03.
The research of the second author was supported by Grant SVV-2011-263316 of the Czech Republic Ministry of Education, Youth and Sports
Communicated by: Julia Knight
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.