Problem 7. Prove, S C X is disconnected if and only if there exists subsets A, B C X such that (i) both A and B are nonempty; (ii) AUB=S; (iii) 203 =...

This problem concerns disconnected metric spaces and subsets

Problem 7.1. Prove, S C X is disconnected if and only if there exists subsets A, B C Xsuch that (i) both A and B are nonempty; (ii) AUB=S;(iii) 203 = (a; and(iv) AflBZU’l. Here the closures are taken with respect to X. [Suggestionz Given A and B, let U = Band V = 2.]