Here length means 1-Hausdorff measure. This seems to be known, what is the reference? Or very short proof?
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asked Mar 19, 2017 at 8:27
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$\begingroup$ What about the standard example of a connected, non-path connected set? Each simplex will have image in one of the two path components, so cohomology will be the direct sum of the two cohomologies which then both are zero as each path component is contractible. $\endgroup$– user1688Commented Mar 19, 2017 at 9:01
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1$\begingroup$ @Antonius Sorry, I do not understand what example you are talking about. For me a standard example is topologist's sine, and what are simplices? $\endgroup$– Fedor PetrovCommented Mar 19, 2017 at 10:35
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$\begingroup$ You need simplices to compute cohomology, that is, singular cohomology. Since your condition is that the first cohomology be finite. $\endgroup$– user1688Commented Mar 19, 2017 at 10:43
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1$\begingroup$ Sorry, maybe I had to specify that $\mathcal{H}^1$ is a Hausdorff measure $\endgroup$– Fedor PetrovCommented Mar 19, 2017 at 10:47
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1$\begingroup$ Theorem 4 of the paper by M H A Newman titled Path Length and Linear measure Proc London Math Soc (3) 2 year 1952 pages 455-468 maybe relevant $\endgroup$– Mohan RamachandranCommented Mar 19, 2017 at 19:33
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1 Answer
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See Exercise 3.5 in "The geometry of fractal sets" by K. J. Falconer.
It says that any such set is an image of rectifiable curve.
For a short proof, check "Rectifiable curve" in my collection.
P.S. The earliest reference I found: Theorem 2 in Continua of finite linear measure. I. by Eilenberg and Harrold (1943).
answered Mar 19, 2017 at 19:57