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.