Exercise 11, § 2

a) Let $X$ be a linearly ordered set with the topology $\mathfrak{T}_0(X)$. If $X$ is connected and has a countable dense subset $A$, show that there exists a homeomorphism (strictly monotonic by virtue of Exercise 8) of $X$ onto one of the intervals of $R$ with end-points 0, 1, which maps $A$ onto the intersection of $Q$ with this interval (use Exercises 9 and 7).

b) Deduce from a) that if $B$ is a subset of $R$ whose complement is dense in $R$, then there exists a homeomorphism of $R$ onto itself which maps $B$ onto a subset of the set $CQ$ of irrational numbers (consider a dense countable subset $A$ of $R$ contained in $CB$).

c) In the set $R \times \{0, 1\}$, linearly ordered by the lexicographic ordering, let $R'$ denote the complement of $Q \times \{1\}$. Show that $R'$, endowed with the topology $\mathfrak{T}_0(R')$, is locally compact and totally disconnected, and contains a countable dense subset $Q'$ homeomorphic to $Q$, but that the topology induced on a non-empty open interval of $R'$ does not have a countable base; in particular such an interval cannot be homeomorphic to any subset of $R$.