Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

24
The categories of this paper are: dg-ga
36
The categories of this paper are: cmp-lg
1
The categories of this paper are: cs
32
The categories of this paper are: atom-ph
10
The categories of this paper are: q-bio
20
The categories of this paper are: acc-phys
27
The categories of this paper are: comp-gas
31
The categories of this paper are: ao-sci
5
The categories of this paper are: gr-qc
6
The categories of this paper are: astro-ph
4
The categories of this paper are: cond-mat
34
The categories of this paper are: chem-ph
25
The categories of this paper are: patt-sol
26
The categories of this paper are: bayes-an
14
The categories of this paper are: math-ph
18
The categories of this paper are: econ