Suppose that (Xi, Pi), i = 1,2,..., k are metric spaces, and form the cartesian product X = Xix X X... X. Define a candidate for a metric on X via p(x, y) = [Pi(x, yi), where = (, ..., xk) and y = (y, ..., yk). (a) Show that p is a metric on X. (b) Suppose that U; (i = 1, ..., k) are subsets of X, open with respect to p. Show that U := U U ... x U is open with respect to p. (c) Formulate an analogous construction of the metric if there are countably infinitely many X. [Warning: the analogue of (b) fails!]