Spanier spaces and the covering theory of non-homotopically path Hausdorff spaces (Q2836436)

We know from Section 2.5 of [\textit{E. H. Spanier}, Algebraic topology. McGraw-Hill Series in Higher Mathematics. New York etc.: McGraw-Hill Book Company. (1966; Zbl 0145.43303)] that if \(\mathcal{U}\) is an open cover of a topological space \(X\) and \(x\in X\), then there is a subgroup \(\pi(\mathcal{U},x)\) of the fundamental group \(\pi(X,x)\) generated by the homotopy classes of closed paths having a representative of the form \(\alpha\beta\alpha^{-1}\), where \(\beta\) is a loop lying in some element of \(\mathcal{U}\) and \(\alpha\) is a path from \(x\) to \(\beta(0)\).NEWLINENEWLINEIn the present paper the group \(\pi(\mathcal{U},x)\) is called \textit{Spanier group with respect to \(\mathcal{U}\)} and the \textit{Spanier group} denoted by \(\pi_1^{sp}(X,x)\) is defined as the intersection of all Spanier groups with respect to all open covers \(\mathcal{U}\). \(X\) is called a \textit{Spanier space} when \(\pi_1^{sp}(X,x)=\pi(\mathcal{U},x)\) for all \(x\in X\).NEWLINENEWLINEFrom the text: ``We study the influence of the Spanier group onNEWLINEthe covering theory and introduce Spanier coverings which are universal coverings in theNEWLINEcategorical sense. Second, we give a necessary and sufficient condition for the existenceNEWLINEof Spanier coverings for non-homotopically path Hausdorff spaces. Finally, we study theNEWLINEtopological properties of Spanier groups and establish criteria for the Hausdorffness ofNEWLINEtopological fundamental groups.''