1. Prove each of the following for a metric space (M, d): (i) The following two statements are equivalent: (a) x € M is not isolated; (b) Every neighborhood of x

contains an infinite number of points of M. (ii) If M has the property that every intersection of open sets is open, then M is discrete. (iii) If M is an infinite metric space, then M contains an infinite open set U such that both U and its complement are infinite.