Covering an uncountable square by countably many continuous functions
Authors:
Wiesław Kubiś and Benjamin Vejnar
Journal:
Proc. Amer. Math. Soc. 140 (2012), 43594368
MSC (2010):
Primary 03E05, 03E15; Secondary 54H05
Published electronically:
May 1, 2012
MathSciNet review:
2957226
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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 nonarchimedean 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/S000299392012112924
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 SVV2011263316 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.
