A topology on the fundamental group.

For an arbitrary topological space algebraic topology prescribes a construction for a fundamental group. There is a natural way of imposing a topology on this set. We will examine this construction and the topological space which results. We use the following notation and definitions: 1. I f X and Y are topological spaces and ACX, BCY and if f:X+Y is a map, we write f: (X,A)➔(Y,B) if f(A)cB. 2. (Y,B)CX,A)={f: (X,A)+(Y,B)I f is continuous}. 3. 1=[0,1], the closed unit interval; i={O,l}, the points O and 1. 4. If X is a topological space and a£X, then R is an equivalence relation on (X,a) ( I ,I ) defined as follows: If f,gE(X,a)(I, i ) then fRg if and only if f and_ g are homotopic relative to I, written f��g rel I. That is, there exists a continuous function F: I xl+X such that F(O,t)= f(t), F(l,t)=g(t), F(x,O)=a, F(x,l)=a for all x,tsl. R is easily shown to be an equivalence relation, [2, p. 6]. We write Rf��{g£(X,a)(I,I )I gRf}. 5. When we speak of a topology on (Y,B) (X,A) we use the "compact-open" topology. The compact-open topology has subbasic sets of the form (C,U) where CCX i�� compact, UC"f is open, and (C,U)��{fE(Y,_B)(X, A) I f(C)cU} . 6. Q(X,a)=(X,a) (I ,I ) with the compact-open topology is called the loop space of X at a.