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

MathSciNet review:
2957226

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Abstract | References | Similar Articles | Additional Information

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 , where is uncountable. This extends Sierpiński's theorem from 1919, saying that can be covered by countably many graphs of functions and inverses of functions if and only if . Using forcing and absoluteness arguments, we also prove the existence of countably many -Lipschitz functions on the Cantor set endowed with the standard non-archimedean metric that cover an uncountable square.

**1.**Uri Abraham and Stefan Geschke,*Covering ℝⁿ⁺¹ by graphs of 𝕟-ary functions and long linear orderings of Turing degrees*, Proc. Amer. Math. Soc.**132**(2004), no. 11, 3367–3377 (electronic). MR**2073314**, 10.1090/S0002-9939-04-07422-2**2.**Uri Abraham, Matatyahu Rubin, and Saharon Shelah,*On the consistency of some partition theorems for continuous colorings, and the structure of ℵ₁-dense real order types*, Ann. Pure Appl. Logic**29**(1985), no. 2, 123–206. MR**801036**, 10.1016/0168-0072(84)90024-1**3.**Stefan Geschke,*A dual open coloring axiom*, Ann. Pure Appl. Logic**140**(2006), no. 1-3, 40–51. MR**2224047**, 10.1016/j.apal.2005.09.003**4.**Thomas Jech,*Set theory*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR**1940513****5.**H. Jerome Keisler,*Logic with the quantifier “there exist uncountably many”*, Ann. Math. Logic**1**(1970), 1–93. MR**0263616****6.**Wiesław Kubiś,*Perfect cliques and 𝐺_{𝛿} colorings of Polish spaces*, Proc. Amer. Math. Soc.**131**(2003), no. 2, 619–623 (electronic). MR**1933354**, 10.1090/S0002-9939-02-06584-X**7.**Wiesław Kubiś and Saharon Shelah,*Analytic colorings*, Ann. Pure Appl. Logic**121**(2003), no. 2-3, 145–161. MR**1982945**, 10.1016/S0168-0072(02)00110-0**8.**Kenneth Kunen,*Set theory*, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR**597342****9.**K. KUNEN,*Forcing and differentiable functions*, preprint, available at http://arxiv.org/abs/0912.3733v2; to appear, DOI:10.1007/s11083-011-9210-8.**10.**Saharon Shelah,*Borel sets with large squares*, Fund. Math.**159**(1999), no. 1, 1–50. MR**1669643****11.**W. SIERPIŃSKI,*Sur un théorème équivalent á l'hypothèse du continu*, Krak. Anz. 1919, 1-3.**12.**W. SIERPIŃSKI,*Sur l'hypothèse du continu*, Fund. Math.**5**(1924) 177-187.**13.**Wacław Sierpiński,*Hypothèse du continu*, Chelsea Publishing Company, New York, N. Y., 1956 (French). 2nd ed. MR**0090558****14.**Piotr Zakrzewski,*On a construction of universally small sets*, Real Anal. Exchange**28**(2002/03), no. 1, 221–226. MR**1973982**

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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

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.