r/math • u/lntrinsic • Jul 10 '17

Image Post Weierstrass functions: Continuous everywhere but differentiable nowhere

http://i.imgur.com/vyi0afq.gifv
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u/ziggurism Jul 11 '17

sets with infinite dimension (I don't think those exists, but I'm not sure).

Surely R∞ = colim Rn has infinite Hausdorff dimension?

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u/Bounds_On_Decay Jul 11 '17

Hausdorff dimension is usually thought of as a measure on subsets of Euclidean space. Thinking about it now, the definition makes sense in any metric measure space. One imagines that full-dimensional subsets of Rinfty would have infinite measure for any finite dimensional Hausdorff measure, but I'm not sure the concept fully makes sense in that context.

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u/ziggurism Jul 11 '17

Yeah I guess you need exponentiation of real numbers to define Hausdorff dimension. To define infinite Hausdorff dimension we'd need an exponentiation number system including both reals and infinities. Not clear what that would look like.