The boundary of an open set as the homotopy limit of the open set minus compact subsets.

Question

Let $M$ be a manifold and $U \subseteq M$ be a relative compact open set of $M$. I run into an equivalence $$\partial U \cong \operatorname{holim}_{K \subseteq U} U \setminus K $$ where the inverse homotopy limit is taken over all compact subsets $K$ of $U$. (One can also think of this as a limit in the $\infty$-category of spaces.) The equality is applied to some carefully constructed $U$'s but there's no condition mentioned so I wonder if this equivalence always holds? If not, is there at least some canonical map from one direction to another?