Exercise 17, § 2

Let $X$ be a connected Hausdorff space such that whenever $X$ is covered by three non-empty connected sets distinct from $X$, there are two of these sets whose union is distinct from $X$

a) Show that, for every point $x \in X$, $C\{x\}$ has at most two components (same method as in Exercise 16).

b) Show that there are at most two points of $X$ whose complements are connected. Moreover, if there are two distinct points $a, b$ with this property, show that, for each $x$ other than $a$ or $b$, $a$ and $b$ belong to different components of $C\{x\}$.

c) Deduce from a) and b) that there is a linear ordering on $X$ compatible with the topology of $X$ (use Exercise 15).