Exercise 18, § 4

a) Show that a compact connected space $X$ cannot be the union of a countable infinity of mutually disjoint non-empty closed sets [by reductio ad absurdum: if $(F_n)$ is a countably infinite partition of $X$ into closed sets, show with the help of Exercise 17 that there is a compact connected set $K$ which does not meet $F_1$ but meets an infinite number of sets $F_n$; in order to do this consider the component of a point of $F_2$ with respect to a compact neighbourhood of $F_2$ not meeting $F_1$. Then use induction].

b) Extend the result of a) to a locally compact connected space $X$ under one or the other of the following two supplementary conditions: one of the $F_n$ is compact and connected, or $X$ is locally connected. [In the first case, reduce to a) by means of Exercise 17].

*c) Let $X$ be the subspace of $\mathbf{R}^3$ which is the union of the following subspaces : $A_n$ is the half-line $x > 0,\ y = 1/n,\ z = 0\ (n \geq 1)$; $B_n$ is the interval $2n < x < 2n + 2,\ y = 0,\ z = 0\ (n \geq 0)$; $C_n$ is the set defined by the relations

where $n \geq 0$.

Show that $X$ is connected and locally compact and is the union of a countable infinite family of mutually disjoint connected non-empty closed sets. \*