习题 18, § 2

a) 令 $X$ 为紧致、连通、局部连通的空间，其在两个点 $a, b$ (章 II, § 4, 习题 19) 之间不可约。 令$x$为$X$中除$a$或$b$之外的点。 证明$C\{x\}$恰好有两个分量，并且$a$和$b$属于是$C\{x\}$的不同分量。 [如习题 16 中的论证，首先证明$C\{x\}$ 不能有两个以上的分量。 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) 令 $X$ 为紧连通空间，其在两个点 $a, b$ 之间不可约。 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)]. \*