Exercise 9, § 2

Let A be an uncountable well-ordered set such that every segment $[u, v]$ is countable.

(a) Prove that the set $L = A\times [0,1[$, totally ordered by the lexicographic order and endowed with the topology $\mathscr{T}_0(L)$ (TG, I, p. 91, exerc. 5), is connected (TG, IV, p. 48, exerc. 7). The space L is called the Alexandroff right. (b) Prove that, for every point $x\in L$, the topological space L$-\{x\}$ is not connected.

(c) Let $\overline{L}$ be the ordered set obtained by adjoining to L points $-\infty$ and $+\infty$ such that $-\infty < x <+\infty$ for every $x\in L$. Endow it with the topology $\mathscr{T}_0(L)$; it is called the completed Alexandroff right. Prove that $\overline{L}$ is compact and connected.

(d) Prove that there does not exist any homeomorphism of $[0,1]$ onto $\overline{L}$ which maps 0 onto $-\infty$ and 1 onto $+\infty$.