Covering an uncountable square by countably many continuous functions

Wiesław Kubiś; Benjamin Vejnar

doi:10.1090/s0002-9939-2012-11292-4

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arXiv · 0710.1402

Covering an uncountable square by countably many continuous functions

Wiesław Kubiś·Benjamin Vejnar

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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 $X$ is an uncountable subset of the real line. This extends Sierpi\'nski'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 the size of $S$ does not exceed $\aleph_1$. Our result is also motivated by Shelah's study of planar Borel sets without perfect rectangles.

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BibTeX RIS

Wiesław Kubiś, Benjamin Vejnar. 2007-10-07. Covering an uncountable square by countably many continuous functions. https://doi.org/10.1090/s0002-9939-2012-11292-4

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