Topological fundamental groupoids: Brown's topology

Abstract

In this paper, we generalize the Brown$^{^,}$s topology on the fundamental groupoids. For a locally path connected space $X$ and a totally disconnected normal subgroupoid $M$ of $\pi X$, we define a topology on the quotient groupoid $\dfrac{\pi X}{M}$ which is a generalization of what introduced by Brown for locally path connected and semilocally simply connected spaces. We prove that $\dfrac{\pi X}{M}$ equipped with this topology is a topological groupoid. Also, we will find a class of subgroupoids of topological groupoids whose their related quotient groupoids will be topological groupoids. By using this, we show that our topology on $\dfrac{\pi X}{M}$ is equivalent to the quotient of the Lasso topology on the topological fundamental groupoids, $\dfrac{\pi^L X}{M}$.