Topological Fundamental Groupoids‎: ‎Brown$^{^,}$s Topology

‎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, ‎$\dfrac{\pi^L X}{M}$ \cite{PS}‎.