What shown below is a reference from "Analysis on manifolds" by James R. Munkres
So Munkres gives a counterexample of open and bounded set that is not rectificable and then as exercise he asks to find a closed and bounded set that is not rectificable. Unfortunately I can't find this such set so I ask to find it. Then I ask if there exist some condictions for which I can claim that a compact set is rectificable. So could someone help me, please?
A fat Cantor set is closed, bounded and of non-zero measure. It has empty interior so every point is a boundary point and every boundary point is in the set (since it is closed).