What are some examples of non-trivial metric spaces that have Hausdorff Dimension of infinity?

I could only think of $\mathbb{R}$ with the discreet metric.


Take the separable Hilbert space of infinite dimension $$ \ell^2=\{(a_n)_{n\in\mathbb{N}}\subseteq \mathbb{R}:\sum_{n\in\mathbb{N}}a_n^2<\infty\} $$

  • 2
    $\begingroup$ +1 Did you know that the subset of $\ell^2$ with all rational coordinates has Hausdorff dimension 1? One of my favorite results of Erdos. $\endgroup$ – Mark McClure Nov 27 '18 at 0:23
  • $\begingroup$ No, I never studied the Hausdorff measure in depth. It's an interesting result, thanks for share it! $\endgroup$ – Dante Grevino Nov 27 '18 at 15:57

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