Talk:Rectifiable set
Add topicLatest comment: 5 years ago by Andylatto in topic Definition of rectifiable sets
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Definition of rectifiable sets
[edit]I'm pretty sure the given definition is wrong. For example, the union of a Cantor subset of with has Hausdorff dimension 1, but is not 1-rectifiable. CBCT (talk) 00:37, 17 April 2020 (UTC)
![{\displaystyle [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d)
![{\displaystyle [2,3]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f35c91443ea609dab5936e902226c21c3224919e)
- Either the definition of purely unrectifiable set given here is incorrect, or the statement that the product of the Smith–Volterra–Cantor set with itself is purely unrectifiable is incorrect. Take the map from R1 to R2 whose image is a horizontal line; the intersection of its image with the square of the Smith–Volterra–Cantor set, when it is nonempty, is just a copy of the Smith–Volterra–Cantor set itself, which has non-zero Hausdorff measure. Unless someone points out an error in this reasoning, or changes the definition given here of purely unrectifiable (which I think is correct), I will change the example to use the standard measure-0 cantor set instead of thhe Smith–Volterra–Cantor set. Andylatto (talk) 06:20, 4 March 2021 (UTC)