Covering maps for locally path-connected spaces (Q2895963)

Quoting the authors, ``this paper is devoted to a theory of coverings by locally path-connected (lpc) spaces.'' An important class of spaces dealt with in this work is that of Peano spaces, i.e., those that are lpc and connected. For a given space \(X\), there is a notion of a universal lpc-space, This consists of an lpc-space \(Y\) and a map \(\pi:Y\to X\) having the unique lifting property for maps of lpc-spaces to \(X\). Such spaces exist for every \(X\) as is shown in Theorem 2.2.NEWLINENEWLINEThe authors note (see after Remark 2.3) that if \(X\) is path-connected, then \(\pi:Y\to X\) as above determines a ``universal Peano space.'' We presume, although it is not stated or proved, that in this case \(Y\) is a Peano space. Of course the meaning of universal in this case is that all maps of Peano spaces to \(X\) have unique lifts.NEWLINENEWLINEThe notion of a Peano map can be found as Definition 6.1. With this, a Peano covering map \(f:X\to Y\) is given in Definition 7.1. It requires:NEWLINENEWLINE1) \(f\) is a Peano map,NEWLINENEWLINE2) \(f\) is a Serre fibration, andNEWLINENEWLINE 3) the fibers of \(f\) have trivial path components.NEWLINENEWLINEWith certain additional conditions (Definition 7.8) such an \(f\) will be called a regular Peano covering map. It turns out that regular Peano covering maps over path-connected spaces are identical to the generalized regular covering maps of Fischer and Zastrow.