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Let $D\subset\mathbb{R}^n$ be a domain and $A\subset D$ such that $\pi_i(A)\subset\mathbb{R}$ is a Lebesgue null set for $1\leq i\leq n$. For $x,y\in D\setminus A$, is there a rectifiable path $p$ connecting x,y? With length $\leq |x-y|+\varepsilon$ ?

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  • $\begingroup$ Is this your homework? $\endgroup$ Nov 14, 2011 at 17:17
  • $\begingroup$ Not only does it look like homework, but it is completely incomprehensible, to boot... $\endgroup$
    – Igor Rivin
    Nov 14, 2011 at 20:40
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    $\begingroup$ It seems clear to me, just a bit laconic as to the style. $\pi_i(A)$ is the projection of $A$ on the $i$-th coordinate (not an homotopy group). $\endgroup$ Nov 14, 2011 at 21:08
  • $\begingroup$ Is $\pi_i(A)$ a projection to $\mathbb{R}^{n-1}$ or to $\mathbb{R}$? The former would suggest (but not prove, IIRC) that $\dim(A)<n-1$, and the latter that $\dim(A)<1$ (where $\dim(A)$ is the Hausdorff dimension). I think it is known that $D\setminus A$ is path-connected if $\dim(A)<n-1$. $\endgroup$ Nov 16, 2011 at 15:54

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