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Covering an uncountable square by countable many continuous functions
- 1.0387008 - MÚ 2013 RIV US eng J - Journal Article
Kubiś, Wieslaw - Vejnar, B.
Covering an uncountable square by countable many continuous functions.
Proceedings of the American Mathematical Society. Roč. 140, č. 12 (2012), s. 4359-4368. ISSN 0002-9939. E-ISSN 1088-6826
R&D Projects: GA AV ČR IAA100190901
Institutional research plan: CEZ:AV0Z10190503
Keywords : uncountable square * covering by continuous functions * set of cardinality N-1
Subject RIV: BA - General Mathematics
Impact factor: 0.609, year: 2012
http://www.ams.org/journals/proc/2012-140-12/S0002-9939-2012-11292-4/home.html
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 × X, where X is uncountable. This extends Sierpiński's theorem from 1919, saying that S × S can be covered by countably many graphs of functions and inverses of functions if and only if |S| <= א 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.
Permanent Link: http://hdl.handle.net/11104/0219404
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