Some Algebraic Properties of Universal and Regular Covering Spaces

Let X̃ be a connected space, X be a space, let p : X̃ −→ X be a continuous map and let (X̃, p) be a covering space of X . In the first section we give some preliminaries from covering spaces and their automorphism groups. In the second section we derive some algebraic properties of both universal and regular covering spaces (X̃, p) of X and also their automorphism groups A(X̃, p). Keywords—covering space, universal covering, regular covering, fundamental group, automorphism group. I. PRELIMINARIES Let X̃ be a connected space, X be a space and let p : X̃ −→ X be a continuous map. If for every x ∈ X has a path connected open neighborhood U such that p−1(U) is open in X̃ and each component of p−1(U) is mapped topologically onto U by p then p is called a covering map (projection). In this case the pair (X̃, p) is called a covering space of X . Let p : E → B and f : X → B be two maps. The lifting problem for f is to determine whether there is a continuous map f ı : X → E