Covering dimension of boundary of compact subset of $\mathbb{R}^n$

Question

Let $X$ be a compact subset of $\mathbb{R}^n$, with the inherited Euclidean topology. Does it follow that $\dim_{cov}(\partial X)\leq n-1$?

I would be happy to have a reference for that, if it is true. Otherwise, I would be happy to get a counter example, or sufficient conditions under which it holds.

Thanks

Answer 1

Theorem 1.8.10 in Engelking's Dimension Theory says that $M \subset \mathbb{R}^n$ has (small inductive) dimension $n$ if and only if $M^\circ \ne \emptyset$.

In separable metric spaces covering dimension is equal to inductive dimension and is monotone w.r.t. inclusion, so this tells us that $\dim M < n$ whenever $M$ has an empty interior. This is always the case for boundaries of closed sets.