Exercise 22, § 1

(a) Let E be an ordered set. A subset U of E is said to be open if, for each $x \in U$, U contains the interval $[x, \rightarrow[$. An open set U is said to be regular if there exists no open set $V \supset U$, distinct from U, such that U is cofinal in V. Show that every open set U is cofinal in exactly one regular open set $\tilde{U}$ [^1]. The mapping $U \rightarrow \tilde{U}$ is increasing. If U, V are two open sets such that $U \cap V = \emptyset$, then also $\tilde{U} \cap \tilde{V} = \emptyset$.

(b) Show that the set $R(E)$ of regular open subsets of E, ordered by inclusion, is a complete Boolean lattice (Exercise 17). For $R(E)$ to consist of two elements it is necessary and sufficient that E should be non-empty and right directed.

(c) If F is a cofinal subset of E, show that the mapping $U \rightarrow U \cap F$ is an isomorphism of $R(E)$ onto $R(F)$.

(d) If $E_1$, $E_2$ are two ordered sets, then every open set in $E_1 \times E_2$ is of the form $U_1 \times U_2$, where $U_i$ is open in $E_i$ ($i=1$, 2). The set $R(E_1 \times E_2)$ is isomorphic to $R(E_1) \times R(E_2)$.

[^1]: This terminology may be justified by noting that there exists a unique topology on E for which the open sets (resp. regular open sets) are those defined in Exercise 22 (a) (cf. General Topology, Chapter I, § 1, Exercise 2 and § 8, Exercise 20).