Recall that an etale topological stack is a stack $\mathscr{X}$ over the category of topological spaces (and open covers) which admits a representable local homeomorphism $X \to \mathscr{X}$ from a topological space. Equivalently, it is a topological stack arising from an etale topological groupoid. It is well known that a *differentiable* stack is etale if and only if all of its automorphism groups are discrete, but the proof involves foliation theory. It seems this proof cannot be extended to the topological setting. However, clearly every etale topological stack has discrete isotropy groups. This begs the question:


**If a topological stack has all of its isotropy groups discrete, is it necessarily etale**?


EDIT: By a topological stack, I mean a stack $\mathscr{X}$ over the category of topological spaces (and open covers) which admits a representable epimorphism $X \to \mathscr{X}$ (not necessarily a local homeomorphism). This is equivalent to saying $\mathscr{X}$ is the stack of torsors for a topological groupoid.


Remark: This question is equivalent to asking if a topological groupoid all of whose isotropy groups are discrete must be Morita equivalent to an etale topological groupoid.



Automorphism groups and etale topological stacks
I don't think this is true. Let $X$ be the quotient of the action of $\mathbb Q$ on $\mathbb R$ by translation. This is a sheaf, and its automorphism groups are trivial. Suppose that there exist a local homeomorphism $U \to X$, where $U$ is non-empty a topological space. Let $V \to U$ be the pullback to $U$ of the $\mathbb Q$-torsor $\mathbb R \to X$; then $V\to \mathbb R$ is a local homeomorphism. By restricting $U$ we may assume that $V = U \times \mathbb Q$; but then $V$ can't be locally connected, and this is a contradiction.


