Exercise 16, § 5

Let $\mathbf{R}_-$ denote the topological space obtained by endowing the linearly ordered set $\mathbf{R}$ of real numbers with the topology $\mathscr{G}_-(\mathbf{R})$ (Chapter I, § 2, Exercise 5). a) Show that every open set in $\mathbf{R}_-$ is the union of a countable family of mutually disjoint intervals which are either open or half-open on the left. Deduce that $\mathbf{R}_-$ is perfectly normal (§ 4, Exercise 8) and totally non-meagre (Exercise 12). b) Show that the space $\mathbf{R}_- \times \mathbf{R}_-$ is not normal, and hence that $\mathbf{R}_-$ is not metrizable, although $\mathbf{R}_-$ contains a countable dense subset. (Considering the set of points $(x, y)$ in $\mathbf{R}_- \times \mathbf{R}_-$ such that $x + y = 1$, show that $\mathbf{R}_- \times \mathbf{R}_-$ has a closed subspace which is homeomorphic to the space $X$ defined in Exercise 15.) c) Show that $\mathbf{R}_-$ is a Lindelöf space (Chapter I, § 9, Exercise 15) and therefore is paracompact (§ 4, Exercise 23). (For each $x \in \mathbf{R}_-$, let $U_x$ be a semi-open interval $]y(x), x]$ with $x$ as right-hand end-point. Show that the set of $z \in \mathbf{R}_-$ which do not belong to any open interval $]y(x), x[$ is countable; use Chapter IV, § 2, Exercise 1.)

d) Show that every compact subset of $\mathbf{R}_-$ is countable [if $A$ is a compact set in $\mathbf{R}_-$, the topologies induced on $A$ by $\mathscr{T}_-(\mathbf{R})$ and $\mathscr{T}_0(\mathbf{R})$ are identical; deduce that every point of $A$ is the left-hand end-point of an interval contiguous to $A$].