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"Normal" geometric shapes have Hausdorff dimensions equal to their topological dimensions. Mandelbrot defined fractals as shapes that have a Hausdorff dimension greater than their topological dimension. Is there a class of shapes that have a Hausdorff dimension less than their topological dimension, or is this impossible? If there is such a shape, what are common examples of them? If this is impossible, why?

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The shapes you are asking about do not exist. The reason is:

Theorem. (Sznirelman) For every metric space $X$, the Hausdorff dimension of $X$ is $\ge$ the covering dimension of $X$.

See for instance section VII.2 of

W.Hurewicz, H.Wallman, Dimension Theory, Princeton University Press.

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    $\begingroup$ I will wear the down-vote on this answer as a badge of honor. $\endgroup$ Apr 8, 2021 at 2:13
  • $\begingroup$ I just saw this (you cited it in a recent comment elsewhere), and thought I'd mention that I've had downvotes on selected answers (more than once), and I think I've even had more downvotes than upvotes on a selected answer, but I don't feel up to looking for any now, and besides, over time I've tended to get upvotes on these "problematic" answers when others have later come across them, so even if I looked now, it's likely that the more extreme such cases are no longer all that extreme. $\endgroup$ Dec 27, 2022 at 17:59

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