## Covering an uncountable square by countably many continuous functions

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- by Wiesław Kubiś and Benjamin Vejnar
- Proc. Amer. Math. Soc.
**140**(2012), 4359-4368 - DOI: https://doi.org/10.1090/S0002-9939-2012-11292-4
- 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 $|S|\leqslant \aleph _1$. 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.## References

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## Bibliographic 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
- 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
- © Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**140**(2012), 4359-4368 - MSC (2010): Primary 03E05, 03E15; Secondary 54H05
- DOI: https://doi.org/10.1090/S0002-9939-2012-11292-4
- MathSciNet review: 2957226