duction to Mathematical Analysis II, Math 302 Homework Assignment #5 - Sections 3.3 - 3.5 1. Let A be a subset of a metric space M . A set U A is open relative to A if U = V A for some open set V M . The notion of a set being closed relative to A is defined similarly. Prove that a set A M is connected if and only if the only subsets of A that are both open relative to A and closed relative to A (at the same time) are and A itself. 2. This is a long problem; it will be worth the same as 2 normal problems. Let F1 be the subset of [0, 1] obtained by removing the middle third, that is \u0014 \u0015 \u0014 \u0015 1 2 F1 = 0, ,1 . 3 3 Let F2 be the subset of F1 obtained by removing the middle third of each interval, that is \u0014 \u0015 \u0014 \u0015 \u0014 \u0015 \u0014 \u0015 1 2 1 2 7 8 F2 = 0, , , ,1 . 9 9 3 3 9 9 In general, Fn is a union of intervals, and Fn+1 is obtained from Fn by removing the middle third of each interval. Let C = n=1 Fn . C is known as the middle thirds Cantor set. Prove that C has the following properties: (a) C is compact. (b) C has infinitely many points. (Hint: Look at the endpoints of Fn .) (c) int(C) = . (d) C is perfect; that is, it is closed with no isolated points. (We defined isolated points on HW #4.) (e) C is totally disconnected; that is, if x, y C with x 6= y, then there exist open sets U and V that disconnect C such that x U and y V