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
(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).© BrainMass Inc. brainmass.com March 4, 2021, 7:52 pm ad1c9bdddf
Complex metric spaces are thoroughly investigated in the solution. The solution is detailed and well presented.