Covering an uncountable square by countably many continuous functions

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F12%3A10103608" target="_blank" >RIV/00216208:11320/12:10103608 - isvavai.cz</a>
Result on the web
<a href="http://www.ams.org/journals/proc/2012-140-12/S0002-9939-2012-11292-4/home.html" target="_blank" >http://www.ams.org/journals/proc/2012-140-12/S0002-9939-2012-11292-4/home.html</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1090/S0002-9939-2012-11292-4" target="_blank" >10.1090/S0002-9939-2012-11292-4</a>

Result language
angličtina
Original language name
Covering an uncountable square by countably many continuous functions
Original language description
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 $Xtimes X$, where $XsubsErr$ is uncountable. This extends Sierpiński''s theorem from 1919, saying that $Stimes S$ can be covered by countably many graphs of functions and inverses of functions if and only if $|S|loealeph_1$. 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.
Czech name
—
Czech description
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Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Project
<a href="/en/project/IAA100190901" target="_blank" >IAA100190901: Topological and geometric structures in Banach spaces</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)<br>S - Specificky vyzkum na vysokych skolach

Publication year
2012
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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