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

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.

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Colby Messinger answered 1 year ago

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