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

DOI:
https://doi.org/10.1090/S0002-9939-2012-11292-4

Published electronically:
May 1, 2012

MathSciNet review:
2957226

Full-text PDF Free Access

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.

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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:
https://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.