Covering an uncountable square by countably many continuous functions

TL;DR

This paper demonstrates that a countable collection of continuous functions can cover an uncountable square in the plane, extending classical results and connecting to Borel set theory.

Contribution

It proves the existence of such a covering for uncountable squares, extending Sierpiński's theorem to continuous functions.

Findings

A countable family of continuous functions covers an uncountable square.

The result generalizes Sierpiński's theorem from 1919.

Connects to Shelah's work on planar Borel sets without perfect rectangles.

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.