On the topology of topological fundamental groupoids

‎This paper is devoted to the Lasso and the Whisker topology of the fundamental groupoid‎. ‎We prove that topological fundamental groupoid of a given space X is locally path connected ‎in‎ ‎general‎ and is path connected if X is simply connected‎. ‎We show that for locally path connected space X‎, ‎the unit map 1 : X → π X is an embedding if and only if X is a semilocally simply connected space‎. ‎Also‎, ‎we give conditions that guarantee Hausdorffness of the topological fundamental groupoid‎.