5. Let (X, d) and (Y,8) be metric spaces and define a metric p on X X Y by, for , (x1, y₁), (x2, y2) EXXY, p((x1, y₁), (x2, y2)) = max{d(x1, x2), 8(y₁, y2)}. (You do not need to verify that p is a metric.) Let (a, b) = X x Y, let U = {xe X: d(a,x) < 1}, and let V {y EY: 8(b, y) < 1}. Prove that U x V is open in the metric topology on X x Y determined by p. = = 5. Let R and RxR have their usual topologies. Let X = (0,27] with the relative topology 1} with the relative topology from from R and let T = {(x, y) € RxR: x² + y² RX R. Define f: XT by f(t) = (cost, sint). Then f is continuous, one-to-one, and onto T. (You need not prove these facts.) Prove that f is not a homeomorphism.