Metric and metrizes the same topology

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1. On R2 let d((x, y), (u, v)) := ((x - u)2 + (y - v)2)1/2 (usual metric), e(x, y) := |x - u|+ |y - v|. Show that e is a metric and metrizes the same topology as d.

2. For any topological space (X, T ) and set A ⊂ X , the boundary of A is defined by ∂ A := A\int A. Show that the boundary of A is closed and is the same as the boundary of X \ A. Show that for any two sets A and B in X,∂( A ∪ B) ⊂ ∂ A ∪ ∂ B. Give an example where ∂( A ∪ B) /= ∂ A ∪ ∂ B.

Reference no: EM13990636

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