MATH 210 Exam 1, Fall 2011

Math 210- Exam 1 Fall 2011 . Show that if U is open in X and A closed in X, then U — A is open in X and A — U is closed in X. . A boundary point to S is a point that is neither interior to S nor interior to S°. (a) Show that zp is a boundary point of S if and only if for every € > 0, N.(z) NS # oF N.(z0) MN S& (b) Let 89 denote the set of all boundary points of S. Show that 0S = 0S° . Show that an upper bound u of a set S is a supremum if and only if for every € > 0 there exists s € S such that u—€ < s. . Show that the finite union of compact sets in a metric space X is compact. Does the same apply on the union of infinitely many compact sets? If not, give a counter example. . Show that the only open and closed subsets of R are ¢ and R. ‘