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 $X\times X$, 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.

💡 Research Summary

The paper investigates the classical problem of covering a Cartesian square by graphs of functions, extending a result originally due to Wacław Sierpiński in 1919. Sierpiński proved that for a subset (S) of the real line, the product (S\times S) can be covered by a countable family of graphs of functions and their inverses if and only if (|S|\le\aleph_{1}). In other words, when the cardinality of (S) exceeds the first uncountable cardinal, no countable collection of (possibly discontinuous) functions suffices.

Motivated by Saharon Shelah’s work on planar Borel sets that contain no perfect rectangle, the authors ask whether the same covering phenomenon can be achieved when the functions are required to be continuous. Their main theorem answers this in the affirmative: for any uncountable set (X\subseteq\mathbb{R}) (including sets of size larger than (\aleph_{1})), there exists a countable family ({f_{n}}_{n\in\mathbb{N}}) of continuous real functions such that

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