Name: \"Cl/V\" \"V V' "' \""J Note: Turn in as many as four solutions, but only the' three highest scores will be considered. GOOD LUCK! 1. Let A be an uncountable set, and let B be a countable set. Show that A is equivalent to AUB- (N0 conta relation between A and B is assumed here) ' E3 Show that the Cantor set A contains no (non-empty) open interval. (Hint: Show that if L .11 E A With then there exists some z E [0, 1] \\ A so that a: [0, 00) be two metrics on M. Prove that the following are equivalent: (a) For each sequence (33,.) in M and :1: E M: :3\" i a: if and only if mu 12) :r. (b) For each subset A of M: A is Open in (M , d) if and only if it is open in (M, p). (B Let (M, d) be a metric space, and let A, B be non-empty subsets of M. Prove or disprove: (1) HA g B, then I g E. (ii) If I g E, then A Q B. (iii) AUB =Zu 6. Let 1 S p 0, the set Aj,r I: {at = (xn) E 2,, : IxJ-l