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Complex Metric Spaces

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Let (S,d) be a metric space and define the function u(x,y) = d(x,y)/(1+d(x,y)) for all x,y

in S.

(a) Prove that u is a metric on S with sup u(x,y) <= 1.

(b) If S = C (complex) and d is the usual Euclidean metric d(z,w) = abs(z-w), then prove that

sup u(z,w) = 1.

(c) For 0 < r < 1, show that u(x,y) < r if and only if d(x,y) < r/(1-r).

(b) Prove that a set is open in (S,u) if and only if it is open in (S,d).

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Complex metric spaces are thoroughly investigated in the solution. The solution is detailed and well presented.

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