Exercise 15, § 6

Let $X$ be the Alexandroff half-line [§ 2, Exercise 12 b)], and let $X'$ be its compactification by adjunction of a point at infinity $\omega$. Show that, for every continuous mapping $f : X \to X$, either we have $\lim_{x \to \omega, x \in X} f(x) = \omega$, or else there exists $z \in X$ such that $f$ is constant in the interval $[z, \to[$. [Show that if, as $x$ tends to $\omega$, $f$ has a cluster point $c \in X$, then we must have $\lim_{x \to \omega, x \in X} = c$; prove that $f$ can have no other cluster point in $X'$, by using the fact that every increasing sequence converges in $X$. Then use Exercise 12 b) of § 2 and the fact that in $X'$ every countable intersection of neighbourhoods of $\omega$ is a neighbourhood of $\omega$].