Topological fundamental groupoid. I

Abstract: We show that the fundamental groupoid~(\Pi_1(X)) of a locally path connected semilocally simply connected space~(X) can be equipped with a \emph{natural} topology so that it becomes a topological groupoid; we also justify the necessity and minimality of these two hypotheses on~(X) in order to topologise the fundamental groupoid. We find that contrary to a belief -- especially among the Operator Algebraists -- the fundamental groupoid is not {\etale}. Further, we prove that the fundamental groupoid of a topological group, in particular a Lie group, is a \emph{transformation groupoid}; again, this result disproves a standard belief that the fundamental groupoids are \emph{far} away from being transformation groupoids. We also discuss the point-set topology on the fundamental groupoid with the intention of making it a locally compact groupoid.