In: Advanced Math
Complete the proof for the claim that any open ball B(x0,r) in Euclidean space Rn is homeomorphic to Rn.
proof is given below the theorem. Show that suggested map g is in fact homeomorphism.
Theorem: Let X0, X1, and X2 be topological spaces and let f: X0 -> X1 and g : X1 -> X2 be continuous functions. Then g∘f : X0 -> X2 is continuous.
proof : Suppose that V is open in X2. Since g is continuous, g-1(V) is open in X1. Since f is continuous, f-1(g-1(V)) = (g∘f)-1(V) is open in X0. It follows that g∘f is continuous.