Analysis II | Part IB, 2002

Suppose that (X,d)(X, d) is a metric space that has the Bolzano-Weierstrass property (that is, any sequence has a convergent subsequence). Let (Y,d)\left(Y, d^{\prime}\right) be any metric space, and suppose that ff is a continuous map of XX onto YY. Show that (Y,d)\left(Y, d^{\prime}\right) also has the Bolzano-Weierstrass property.

Show also that if ff is a bijection of XX onto YY, then f1:YXf^{-1}: Y \rightarrow X is continuous.

By considering the map xeixx \mapsto e^{i x} defined on the real interval [π/2,π/2][-\pi / 2, \pi / 2], or otherwise, show that there exists a continuous choice of arg zz for the complex number zz lying in the right half-plane {x+iy:x>0}\{x+i y: x>0\}.

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