Exercise 16, § 2

Let $X$ be a connected Hausdorff space which is completely irreducible between two distinct points $a$ and $b$, that is to say such that there is no connected subset of $X$ other than $X$ itself which contains both $a$ and $b$. Show that $X' = C\{a, b\}$ is connected and that, for each $x \in X'$, the complement of $x$ in $X$ has exactly two components $A(x)$ and $B(x)$ such that $a \in A(x), b \in B(x), a \notin B(x), b \notin A(x)$. [Show that $C\{x\}$ cannot be partitioned into three non-empty open sets, by using Exercise 4 a) of Chapter I, § 11.] Deduce that there exists a linear ordering on $X$ which is compatible with the topology of $X$ (use Exercise 15).