1. The problem statement, all variables and given/known data Let X be set donoted by the discrete metrics d(x; y) =(0 if x = y; 1 if x not equal y: (a) Show that any sub set Y of X is open in X (b) Show that any sub set Y of X is closed in y 2. Relevant equations In a topological space, a set is closed if and only if it coincides with its closure. Equivalently, a set is closed if and only if it contains all of its limit points. A subset U of a metric space (M, d) is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in M with d(x, y) < ε, y also belongs to U. Equivalently, U is open if every point in U has a neighbourhood contained in U. 3. The attempt at a solution I tried to use the line y=x and said that there is a finite many balls centered along the line, that means its open because of the union. But my prof said that we r working in abstract so we cant use R2. How would this work then?