5. A topological group, G, is a Hausdorff topological space such that the group operation : (1,y) Tuy from G x G into G is continuous. We will simply write xy instead of ey. We will denote the identity of G by e. Prove the following claims. (a) (5 points) Let a and 8 are two paths in G the function a.3:1 + G defined by (a 8) (t) = a(t) 8 (t) is continuous, hence a path in G. (Hint: Use the composition 1 GG G where * (t) = (a (t), 8(t)). (b) (5 points) If 0,8,9,8 P(G,e,e), a Brel {0,1), and y8 rel {0,1}, then ay 3.8 rel {0,1}. (Hint: If F:a Brel 0,1} and G:98 rel {0,1}, then consider F.G.) (c) (5 points) Let a, 8 P(G,e,e), then (a B] (a * 8]. (Hint: a ~a* Ex. rel {0,1} and 8. * 8 rel {0,1). Now use (b).) (d) (5 points) Show that # (G,e) is abelian. (Hint: a 2500 *a rel {0,1} and B B*Ex, rel {0,1}, Now use (6) and (c).)