Let $X$ be a locally compact, Hausdorff, second countable space. Let $(U_k)_{k\in\mathbb{N}}$ be a countable family of closed subsets in $X$ with $\mathrm{dim}(\partial_X U_k)\leq n$, for all $k$ (where $\mathrm{dim}$ stands for topological covering dimension). Does it follow that $\mathrm{dim}(\partial_X (\bigcap_{k\in\mathbb{N}} U_k))\leq n ?$

If the collection was finite, then the answer would be 'yes', as $\partial_X (\bigcap\limits_{k=1}^{m} U_k)\subseteq \bigcup\limits_{k=1}^{m} \partial_X U_k$.

It is not clear that a similar inclusion holds for infinite collections, as here Boundary, unions and intersections I think that there is a mistake in the answer, the inclusion $\mathrm{int}( \bigcap \limits_{A \in \mathscr{A}} A)\subseteq \bigcap \limits_{A \in \mathscr{A}} \mathrm{int}(A)$ is reversed.

If the answer is 'no', I would be still happy to know about a class of topological spaces in which this holds (namely, having more restrictions on $X$).


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