A covering morphism \(p: \tilde{X}\rightarrow X\) is called universal if it covers every cover of \(X\) which means that for any other covering morphism of \(X\), \(q:\tilde{Y}\rightarrow X \), there exists a continuous function \(f\colon \tilde{X}\rightarrow \tilde{Y}\) such that \(p=qf\). If a space \(X\) is connected, locally path connected and semi-locally simply connected, then it has a universal cover as stated in [R. Brown, Topology and groupoids. 3rd revised, updated and extended ed. Bangor: Ronald Brown (2006; Zbl 1093.55001)].
In this paper, the authors prove the existence of universal covering maps by open covers of a space \(X\) with the help of Spanier groups defined in [E. H. Spanier, Algebraic topology. New York etc.: McGraw-Hill Book Company (1966; Zbl 0145.43303)] and named in [H. Fischer et al., Topology Appl. 158, No. 3, 397–408 (2011; Zbl 1219.54028)]. The purpose of using Spanier groups is to change the condition of semi-locally simply connectedness. Then the authors show that all universal coverings \(p: \tilde{X}\rightarrow X\) are Spanier coverings as defined in [B. Mashayekhy et al., Georgian Math. J. 20, No. 2, 303–317 (2013; Zbl 1276.57005)], i.e. coverings such that \(\tilde{X}\) is a Spanier space. In addition, they study the conditions under which a one point union of spaces has a universal covering.