The fundamental groupoid as a topological groupoid: Lasso topology (Q2230903)

This paper is concerned with the topology of the fundamental groupoid \(\pi X\) of a locally path connected topological space \(X\). Ultimately, the authors define a basis for a topology on \(\pi X\) for which it is a topological groupoid (Theorem 2.5). This construction defines a functor from the category of (locally path connected) topological spaces to the category of topological groupoids (Proposition 2.6 and Corollary 2.7). Recall that the fundamental groupoid \(\pi X\) of a topological space \(X\) is defined to be the category whose objects are the points of \(X\), and whose arrows are homotopy classes of paths with fixed endpoints. This construction simultaneously generalizes the universal covering space and the fundamental group of \(X\) in that, given \(x\in X\), the set of all arrows starting at \(x\) coincides with the universal covering \(\tilde{X}\) and the isotropy group at \(x\) (referred to also as ``object group'') is precisely \(\pi_1(X,x)\). In [\textit{Ž. Virk} and \textit{A. Zastrow}, Topology Appl. 231, 186--196 (2017; Zbl 1385.57002)], building upon the theory of [\textit{N. Brodskiy} et al., Fundam. Math. 218, No. 1, 13--46 (2012; Zbl 1260.55013)], the authors prove that there exists a topology on \(\tilde{X}\) extending the so-called Lasso topology on the fundamental group, and pose the question of whether this topology can be further extended to the entire fundamental groupoid. In the present article, the authors prove that this is indeed the case. Very briefly, the necessary notions to define the Lasso topology are recalled, the basis is defined and proved to be so (Proposition 2.4), and the main results are proven. Then, it is shown that this topology indeed restricts to the Lasso topologies on \(\tilde{X}\) and \(\pi_1(X,x)\) (Proposition 2.10 and Corollary 2.11). As an application, the article concludes by proving that \(X\) is semi-locally simply connected if and only if the sets of arrows with fixed source and target are discrete in the larger Lasso topology (Proposition 2.12). Although the proofs are very clear, the article is at times obscured by the presentation and the broken English. Aside from a couple of redundancies that can be brushed over, some statements could be regarded as imprecise, e.g., in Corollary 2.7, it is claimed that there is a functor from the full category of topological spaces when all previous results are valid only for locally path connected spaces. No further comment is presented in this regard. One typo in the definition of the ``Spanier group'' with respect to the open cover \(\mathcal{U}\): ``\(\pi(\mathcal{U},x)\)'' should read ``\(\pi_1(\mathcal{U},x)\)''.