2
$\begingroup$

Let $A\subset \mathbb{R}^n$ such that $\pi_i(A)$ is a Lebesgue null set for $1\leq i\leq n$. Is the complement $\mathbb{R}^n\!\setminus\! A$ pathwise connected? If so, is it always possible to construct a rectifiable path from $x$ to $y$ in the complement with length near $|x-y|$ ?

  • 0
    The solution to the other problem was very helpful. It would be great to also see how path length $|x-y|_2$ (or $\leq |x-y|_2+\varepsilon$ for arbitrary \varepsilon>0) could be achieved.2011-11-16

0 Answers 0