It should suffice that I get a hint; I would just like to know what particular characteristic of $\Bbb S^1$ makes it have a Hausdorff dimension $1$. Thanks.
How to prove that the unit sphere $\Bbb S^1$ has Hausdorff dimension $1$?
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measure-theory
dimension-theory
hausdorff-measure
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1Do you know how to prove that the unit interval has Hausdorff dimension 1? – 2017-02-10
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0The justification is similar to that of the Hausdorff dimension of $\mathbb{R}$ that is 1. The d-dimensional Hausdorff content is equal to $0$ for $d\geq 1$, so when taking the infimum to calculate $dim_H$ you get $1$. – 2017-02-10
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0@Test123 I'm not quite sure how to prove that the $a-$dimensional Hausdorff content is $>0$ for $a < 1$. – 2017-02-10
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0@MoisheCohen no, but I'll give it a try. – 2017-02-10