Exercise 4, § 11

Let $X$ be a connected space of at least two points.

a) Let $A$ be a connected subset of $X$, $B$ a subset of $\mathbf{C}A$ which is both open and closed in $\mathbf{C}A$. Show that $A \cup B$ is connected. (Apply Exercise 5 of § 3 to $Y = A \cup B$ and $Z = \mathbf{C}A$.)

b) Let $A$ be a connected subset of $X$ and $B$ a component of the set $\mathbf{C}A$. Show that $\mathbf{C}B$ is connected [use a)].

c) Deduce from b) that there exist two connected subsets $M, N$ of $X$ which are distinct from $X$ and such that $M \cup N = X$ and $M \cap N = \varnothing$.