[1]:http://groupoids.org.uk/topgpds.html

Part  of 10.5.8 of [Topology and Groupoids][1] is, in a more usual notation, essentially  the following, in which $\sigma, \tau$ are the source and target maps, $St_G x$ is $\sigma ^{-1} x$,    by $N$ is _totally disconnected_ is meant that $N(x,y)$ is empty for $x \ne y$, and _normality_ of $N$ in $G$ also means that $N,G$ have the same set of objects: 

 Let $X$  be a  space which admits a universal cover, and let $N$ be a totally disconnected normal subgroupoid of the fundamental groupoid $\pi_1( X) $,  Then the set of elements of the
quotient groupoid $\pi_1(X)/N$ may be given a topology such that the projection
$$q = (\sigma, \tau) : \pi_1(X)/N \to  X \times X$$ is a covering map and for $x \in  X$  the target map $\tau :St_{\pi_1(  X)/N} \to X$ is the covering map determined by the normal subgroup $N(x)$ of $\pi_1(X, x)$.

So this uses all the points of $X$ and puts all these covers into a covering space, which means you don't make a choice of base point;  instead you use all the choices. Further, $\pi_1(X)/N$ with this topology is actually a topological groupoid. 

This may be the optimal  way of answering the question.