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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 $X\times X$, where $X\subs\Err$ is uncountable. This extends Sierpiński''s theorem from 1919, saying that $S\times S$ can be covered by countably many graphs of functions and inverses of functions if and only if $|S|\loe\aleph_1$. 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.
  • 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 $X\times X$, where $X\subs\Err$ is uncountable. This extends Sierpiński''s theorem from 1919, saying that $S\times S$ can be covered by countably many graphs of functions and inverses of functions if and only if $|S|\loe\aleph_1$. 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. (en)
Title
  • Covering an uncountable square by countably many continuous functions
  • Covering an uncountable square by countably many continuous functions (en)
skos:prefLabel
  • Covering an uncountable square by countably many continuous functions
  • Covering an uncountable square by countably many continuous functions (en)
skos:notation
  • RIV/00216208:11320/12:10103608!RIV13-AV0-11320___
http://linked.open...avai/predkladatel
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  • P(IAA100190901), S, Z(AV0Z10190503)
http://linked.open...iv/cisloPeriodika
  • 5
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  • 128953
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  • RIV/00216208:11320/12:10103608
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  • set of cardinality $\aleph_1$; covering by continuous functions; Uncountable square (en)
http://linked.open.../riv/klicoveSlovo
http://linked.open...odStatuVydavatele
  • US - Spojené státy americké
http://linked.open...ontrolniKodProRIV
  • [C97C3A04C7DC]
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  • Proceedings of the American Mathematical Society
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  • 140
http://linked.open...iv/tvurceVysledku
  • Kubiś, Wieslaw
  • Vejnar, Benjamin
http://linked.open...ain/vavai/riv/wos
  • 000312117500033
http://linked.open...n/vavai/riv/zamer
issn
  • 0002-9939
number of pages
http://bibframe.org/vocab/doi
  • 10.1090/S0002-9939-2012-11292-4
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  • 11320
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