Exercise 12, § 2

a) Let $A$ be a well-ordered set and let $I$ denote the interval [0, 1[ of $R$. Show that the set $X = A \times I$, linearly ordered by the lexicographic ordering and endowed with the topology $\mathfrak{T}_0(X)$, is connected (Exercise 7); there are therefore linearly ordered sets $X$ which are connected in the topology $\mathfrak{T}_0(X)$ and have an arbitrary cardinal [cf. , Exercise 7 b)]

b) Take $A$ to be an uncountable well-ordered set in which every segment ]$\leftarrow, t$ is countable : the corresponding topological space $X$ is called the Alexandroff half-line. Show that for every interval $[a, b]$ of $X$ ($a < b$) there exists a strictly increasing homeomorphism of $[a, b]$ onto the interval $[0, 1]$ of $\mathbf{R}$. (Prove by transfinite induction that there exists a strictly increasing homeomorphism of $A \cap [a, b]$, $A$ being identified with the set of points $(t, 0)$ of $X$ onto a subspace of $[0, 1]$.)