Covering an uncountable square by countably many continuous functions

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 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 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.

Added new results (9 pages)

References in corpus (1)

• Borel sets with large squares

Cited by in corpus (2)

• Packing index of subsets in Polish groups
• Forcing Differentiable Functions