Exercise 8, § 2

Let $X$ and $Y$ be two linearly ordered sets, endowed with the topologies $\mathfrak{T}_0(X)$ and $\mathfrak{T}_0(Y)$ respectively.

a) Suppose $X$ connected and let $f : X \to Y$ be a continuous mapping. Show that, for all $x, y$ in $X$ such that $x < y$, every $z \in Y$ belonging to the closed interval with end-points $f(x)$ and $f(y)$ belongs to $f(X)$ (use Exercise 7). Deduce that $f$ is a homeomorphism of $X$ onto $f(X)$ if and only if $f$ is continuous and strictly monotone.

b) Give an example of a homeomorphism of the rational line $Q$ onto itself which is not strictly monotone.