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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 ...

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}(...