COVERING AN UNCOUNTABLE SQUARE BY COUNTABLY MANY CONTINUOUS FUNCTIONS.

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 × X, where X ⊆ … is uncountable. This extends Sierpiński's theorem from 1919, saying that S × S can be covered by countably many graphs of functions and inverses of functions if and only if |S| ≤ …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. [ABSTRACT FROM AUTHOR]