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Previously asked at MSE: Let $\mathscr{H}$ be the space of compact nonempty subsets of $\mathbb{R}^2$ (I'm not especially wedded to dimension $2$, so feel free to tweak that if it would lead to a more ...

Let $(X,d)$ be a compact metric space and recall that the $\epsilon$-external covering number $\mathcal{N}^{\epsilon}(X)$ of $X$ is defined by: $$ \mathcal{N}^{\epsilon}(X) := \inf\left\{ N\in \mathbb{...

In the same way that we can say manifolds are made of pieces that look like $\mathbb{R}^n$, is there any way to say that spaces with the same hausdorff dimension are made up of pieces that look the ...

This question is inspired from the definition of Hausdorff measure. Let $C$ be a closed unit hypercube in $\mathbb R^d$ (side length equal to one, including boundary. The cube itself is at top ...

I am thinking of the following question: Let $X\subseteq \mathbb R$. Is it true that $$ \mathrm{dim_H}(X\times X)=2\mathrm{dim_H}(X)? $$ My thoughts: We know that $\mathrm{dim_H}(X)+\mathrm{dim_H}(...