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Complete the proof for the claim that any open ball B(x0,r) in Euclidean space Rn is...

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Advance Math

Complete the proof for the claim that any open ballB(x0,r) in Euclidean space Rn is homeomorphicto Rn.

proof is given below the theorem. Show that suggested map g isin fact homeomorphism.

Theorem: Let X0, X1, and X2 betopological spaces and let f: X0 -> X1 andg : X1 -> X2 be continuous functions. Theng?f : X0 -> X2 is continuous.

proof : Suppose that V is open in X2. Since g iscontinuous, g-1(V) is open in X1. Since f iscontinuous, f-1(g-1(V)) =(g?f)-1(V) is open in X0. It follows that g?fis continuous.

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