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On the Universal Covering Space and the Fundamental Group

1953; American Mathematical Society; Volume: 4; Issue: 4 Linguagem: Inglês

10.2307/2032541

1088-6826

C. W. Saalfrank,

Fuzzy and Soft Set Theory

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).1. Lemma 1.1.If A is a retract of X and pEA, then Ap* is homeomorphic to a subset of Xp*.Proof.Consider any a*EAp* and any way aEa*-Then a:(0, 1) -*A.Define a': (0, 1)-*X by a'\t) =a(i) for all iG(0, 1), and let x* be the element of X* such that a'Ex*.Defining A(o*) =x*, it is easily shown that A is single-valued.To show that A is continuous, we consider any neighborhood U*iU, a) of x* where A(a*) =x*.Then U*iU, a) consists of all g* in X* such that g*Daß with ßit)CUfor all tEiO, 1), and where ctEx* and U is any neighborhood in X of a(l).Consider any 7Go*.Then y'Ex* and therefore y'~a.Hence there exists a continuous map A':(0, 1)X(0, l)-+X such that k' satisfies the E-conditions for (7', a, X).We define A:(0, 1)X(0, l)-+A by kit, s)=rk'it, s).It follows easily that ra(0)=7(0), ra(l)=7(l), and that A satisfies the E-conditions for ira, 7, A).Hence ra~7 and we have raEa*-Take the neighborhood F*(Z/f\4, rat) of a*.Now consider any ai* EV*iUr\A, roc) and let A(ai*)=xi*.Then ai* contains a way (ra)/S