Exercise 16, § 9

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

a) The set of condensation points of a subset $A$ of $X$ is closed. Give an example in which $X$ has only one condensation point (cf. no. 8, Theorem 4).

b) Suppose $X$ is an accessible space in which every open subspace is Lindelöf (Exercise 15). Show that for every subset $A$ of $X$, the set $B$ of condensation points of $A$ is perfect, and that $A \cap \complement B$ is countable.