Exercise 19, § 2

a) Let $X$ be a connected Hausdorff space and let $a, b$ be two distinct points of $X$ such that there is no connected \emph{closed} subset of $X$, other than $X$ itself, which contains both $a$ and $b$. If $x$ is a point of $X$ other than $a$ or $b$, then $C\{x\}$ is not connected if and only if, for each $y \in C\{x\}$, there is in $C\{y\}$ a connected subset which is closed in $X$ and contains $x$ and one of the points $a, b$. [To show that the condition is necessary, use Exercise 4 a) of Chapter I, § 11; to show that it is sufficient, let $A$ (resp. $B$) be the set of all $y \in X$ such that there is a connected subset of $C\{y\}$ which is closed in $X$ and contains $a$ and $x$ (resp. $b$ and $x$); show that $A$ and $B$ are non-empty and open and disjoint from each other, and that $C\{x\} = A \cup B$.

b) Let $X$ be a compact connected space which is irreducible between two of its points $a, b$ (Chapter II, § 4, Exercise 19). Show that if for each pair of distinct points $x, y$ of $X$ there is a compact connected set in $X$ which contains $y$ and one of $a, b$ but does not contain $x$, then there is a linear ordering on $X$ such that $\mathfrak{T}_0(X)$ is the given topology on $X$ (Exercises 15 and 16).