Covering an uncountable square by countable many continuous functions

0387008 - MÚ 2013 RIV US eng J - Článek v odborném periodiku
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
Grant CEP: GA AV ČR IAA100190901
Výzkumný záměr: CEZ:AV0Z10190503
Klíčová slova: uncountable square * covering by continuous functions * set of cardinality N-1
Kód oboru RIV: BA - Obecná matematika
Impakt faktor: 0.609, rok: 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.
Trvalý link: http://hdl.handle.net/11104/0219404