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So, this a question that arose from reading the first few pages of Ramanan's (excellent) book "Global Analysis", which uses modern algebraic notions used mainly in Category theory, while avoiding any actual Category theory.

Let $X$ be some topological space, and let $\mathcal F$ be some presheaf defined on $X$. Let us assume $X$ fulfills some topological axioms, say it is Hausdorff. Is there some necessary condition that $\mathcal F$ must fulfill in order to make sure that the associated etale space (IE the disjoint union of all stalks $\mathcal F _x$ for all $x\in X$ together with the topology generated by the sets $\tilde s (U)$, the germs of elements of the open sets of $X$) is also Hausdorff? Is it Hausdorff for any presheaf? What about other topological axioms?

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    $\begingroup$ Certainly it's going to be a rare property. The étale space of continuous real-valued functions tends to be non-Hausdorff. $\endgroup$ – Sofie Verbeek Sep 12 '18 at 15:16
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    $\begingroup$ Unravelling the definitions to show what it means for an étale space to be Hausdorff gives you a rather weird (tautological) criterion. On first sight it does not seem to have a decent intuitive interpretation. $\endgroup$ – Sofie Verbeek Sep 12 '18 at 15:19

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