Exercise 23, § 1

Let E be an ordered set and let $R_0(E) = R(E) - \{\emptyset\}$ (Exercise 22). For each $x \in E$, let $r(x)$ denote the unique regular open set in which the interval $[x, \rightarrow[$ (which is an open set) is cofinal. The mapping $r$ so defined is called the canonical mapping of E into $R_0(E)$. Endow $R_0(E)$ with the order relation opposite to the relation of inclusion.

(a) Show that the mapping $r$ is increasing and that $r(E)$ is cofinal in $R_0(E)$.

(b) An ordered set E is said to be antidirected if the canonical mapping $r : E \rightarrow R_0(E)$ is injective. For this to be so it is necessary and sufficient that the following two conditions should be satisfied :

(I) If $x$ and $y$ are two elements of E such that $x < y$, there exists $z \in E$ such that $x < z$ and such that the intervals $[y, \rightarrow[$ and $[z, \rightarrow[$ do not intersect.

(II) If $x$ and $y$ are two non-comparable elements of E, then either there exists $x' \geqslant x$ such that the intervals $[x', \rightarrow[$ and $[y, \rightarrow[$ do not intersect, or else there exists $y' \geqslant y$ such that the intervals $[x, \rightarrow[$ and $[y', \rightarrow[$ do not intersect.

(c) Show that, for every ordered set E, $R_0(E)$ is antidirected and that the canonical mapping of $R_0(E)$ into $R_0(R_0(E))$ is bijective (use Exercise 22 (a)).