pps assist to answer question 1

As background reading I recommend Rudin "Principles of Mathematical Analysis" (The Metric Spaces part in Chapter 2) as well as the according Chapters in Aliprantis and Border "Infinite Dimensional Analysis". (1) Consider the ball Bi (x) = {y ER' | d(x, y) } such that . = (0,0) and A = 1. Please draw the respective balls for the following two metrics on R2: (a) d(r, y) = max {lr1 - yil . (x2 - 921} (supremum norm) (b) d(x, y) = 1 for a # y ( discrete metric) (2) Let X be the closed unit interval [0, 1]. Consider the Euclidean metric space ([0, 1] , d) with distance function d (x, y) = |x - yl for all a, y E X. Specify for the following subsets of X whether they are conver as well as closed or / and open in this Euclidean topology: (a) e (b) [0, 1] (c) {} for some n E N (d) U{} TEN