3 Suppose that (Xi,i),i=1,2,,k are metric spaces, and form the cartesian product X=X1 X2Xk. Define a candidate for a metric on X via (x,y)=i(xi,yi), where x=(x1,,xk) and y=(y1,,yk) (a) Show that is a metric on X. (b) Suppose that Ui(i=1,,k) are subsets of Xi open with respect to i. Show that U:=U1U2 Uk is open with respect to . (c) Formulate an analogous construction of the metric if there are countably infinitely many Xi. [Warning: the analogue of (b) fails!]