When one defines the etale space of a presheaf $\mathscr F$ on a topological space $X$, would be assumed that $X$ is a $T0$-space (i.e. for every $x$, $y$ in $X$ exists an open set containg one of them but not the other point)?
If $X$ is not $T0$, I'm not sure that the stalks of $\mathscr F_x$ are disjoint each other for all $x\in X$.