Let $X$ be a topological space and let $S \subset X$ be a subspace with induced topology (not necessarily open or closed). Let $i : S \to X$ be the inclusion map.

Assume moreover that for any subspace $K \subset X$ there exists a decreasing family of open subsets $(V_i)_{i \in I}$ of $X$ such that $\lim_{i \in I} V_i = K$ (as is for example the case when working with locally profinite spaces).

Moreover let $\mathcal{F}$ be a sheaf over $X$.

As already known in sheaf theory, one can construct a "pullback presheaf" as follows: For $U \subset S$ open, one can define the presheaf $i^{-1}\mathcal{F}(U) = \varinjlim_{V \supset U} \mathcal{F}(V)$. The pullback sheaf $i^* \mathcal{F}$ is simply sheaving the pullback presheaf.

My question: In the case of inclusions, do $i^* \mathcal{F}$ and $i^{-1} \mathcal{F}$ coincide?


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