Can the set of Real numbers be partioned into two parts such that both are uncountable,dense and have empty interior and any closed interval intersects both at uncountably many points?
$\begingroup$
$\endgroup$
0
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
Yes. Let $A$ be the set of reals whose decimal expansion has infinitely many occurrences of 7, and let $B$ be the set of reals whose expansions have only finitely many such occurrences.
These sets partition $\mathbb{R}$. They are both uncountable, since I can vary the digits between or beyond the 7's at will. They are both dense, because I can start with any given finite sequence before deciding whether to have infinitely many 7's or not. They each have empty interior, since every interval has uncountably many real numbers of each kind.
answered Apr 13, 2016 at 11:14