Covering an uncountable square by countably many continuous functions

Details Covering an uncountable square by countably many continuous functions Covering an uncountable square by countably many continuous functions

2007-10-07

arxiv.org/abs/0710.1402v3

Mathematics

Logic

Added new results (9 pages)

Scientific paper

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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