Exercise 18, § 2

a) Let $X$ be a compact, connected, locally connected space which is irreducible between two of its points $a, b$ (Chapter II, § 4, Exercise 19). Let $x$ be a point of $X$ other than $a$ or $b$. Show that $C\{x\}$ has exactly two components and that $a$ and $b$ belong to different components of $C\{x\}$. [Arguing as in Exercise 16, show first that $C\{x\}$ cannot have more than two components. Next, if $V$ is any connected closed neighbourhood of $x$ which contains neither $a$ nor $b$, let $A_V, B_V$ be the components of $a, b$ respectively in $C_V$; show that $A_V \neq B_V$ by using Exercise 17 of Chapter II, § 4; finally consider the union of the sets $A_V$ (resp. $B_V$) as $V$ runs through the set of all connected closed neighbourhoods of $x$ in $X$.] Deduce that there exists a linear ordering on $X$ such that $\mathfrak{T}_0(X)$ is the given topology on $X$ [Exercises 15 and 14 c)].

b) Let $X$ be a compact connected space which is irreducible between two of its points $a, b$. Show that if $X$ has more than one prime constituent (Chapter II, § 4, Exercise 20) then the space $X'$ of prime constituents of $X$ (\emph{loc. cit.}) is irreducible between the prime constituents of $a$ and $b$, and that there is a linear ordering on $X'$ such that the topology of $X'$ is identical with $\mathfrak{T}_0(X')$. \* Give an example in which $X'$ is distinct from $X$ [cf. Exercise 15 d)]. \*