The issues that arise when moving from topological groups to topological groupoids are (at least to me) both subtle and interesting. Recently, I was reading a paper of R. Brown and J.P.L. Hardy from 1974 which lists the following as an open problem:
Problem: Find a topological groupoid $G$ and a totally disconnected normal subgroupoid $N$ such that $G\rightarrow G/N$ (the map of morphism spaces) is not an open map.
Of course, this is not an issue with topological groups and normal subgroups.
Has this problem been settled? It has been a while and seems pretty accessible so I would imagine some progress has been made. If such $G$ and $N$ do exist, are there reasonable conditions we can place on them that do make $G\rightarrow G/N$ an open map?
I will make a couple of clarifications based on David's comment.
Clarification: Totally disconnected in this context is not topological. It means that $N(x,y)=\emptyset$ when $x\neq y$ so the components of $N$ (as a groupoid, not a space) are groups. R. Brown talks about these and quotient groupoids (but not topological groupoids) in "Topology and Groupoids."
As David points out, the problem refers to whether or not the morphism component of the canonical functor $G\rightarrow G/N$ is an open map of topological spaces.