On the Universal Covering Space and the Fundamental Group

In this note it will be shown that when A is a retract (cf. [3])1 of X, then the same relationship holds between the universal covering spaces of A and X (Theorem 1.1) and also between the fundamental groups of A and X (Theorem 2.2). We denote the universal covering space of X by X*, where pEX, and the fundamental group by iriX) [l]. All spaces are assumed to be arcwise connected, Hausdorff spaces. If a and ß are continuous maps of the closed unit interval (0, 1) into X, then we use the notation a~ß (equivalent) to mean: a(0)=/3(0), a(l)=j3(l), and there exists a continuous map kit, s) such that (i) A:(0, 1)X(0, 1)-** (into), (ii) kit, 0) =ait), kit, 1) =ßit) for all i€(0, 1), (iii) A(0, s)=a(0)=j3(0), A(l, s) =a(l) =/3(l) for all sEiO, 1). A continuous map k satisfying the above conditions i, ii, and ¡ii will be said to satisfy the E-conditions for (a, ß, X).