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 $X\subs\Err$ 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.

Keywords

• Uncountable square
• covering by continuous functions
• set of cardinality $\aleph_1$