Covering an uncountable square by countably many continuous functions

Kubiś, Wiesław; Vejnar, Benjamin

doi:10.1090/S0002-9939-2012-11292-4

Covering an uncountable square by countably many continuous functions
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by Wiesław Kubiś and Benjamin Vejnar

Proc. Amer. Math. Soc. 140 (2012), 4359-4368

DOI: https://doi.org/10.1090/S0002-9939-2012-11292-4

Published electronically: May 1, 2012

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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 $X\times X$, where $X\subseteq \mathbb {R}$ 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 $|S|\leqslant \aleph _1$. 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.

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