Definition. A metric space (X, d) is said to be perfect when it satisfies the following conditions: (i) (X, d) is complete, and (ii) there is no a X which is an isolated point. (Recall: the notion of isolated point a X was introduced in Lecture 1, cf. Definition 1.15.) Problem P11.4 Let (X, d) be a perfect metric space. Prove that the set X is uncountable