Exercise 17, § 9

Let $X$ be a topological space, $A$ a subset of $X$. A point $a \in X$ is said to be a condensation point of A if in every neighbourhood of $a$, there exists an uncountable infinity of points of A.

a) The set of condensation points of a subset A of X is closed. Give an example where X has only one condensation point (cf. I, p. 67, th. 4).

b) Suppose that X is an accessible space in which every open subspace is a Lindelöf space (exerc. 16). Now prove that for every subset A of X, the set B of condensation points of A is perfect, and the set $A \cap \complement B$ is countable.