Separated Subsets of Linearly Ordered Space under Order Topology

Theorem
Let $T = \left({S, \preceq, \tau}\right)$ be a linearly ordered space.

Let $A$ and $B$ be separated sets of $T$.

Let $A^*$ and $B^*$ be defined as:
 * $A^* := \displaystyle \bigcup \left\{ {\left[{a \,.\,.\, b}\right]: a, b \in A, \left[{a \,.\,.\, b}\right] \cap B^- = \varnothing}\right\}$
 * $B^* := \displaystyle \bigcup \left\{ {\left[{a \,.\,.\, b}\right]: a, b \in B, \left[{a \,.\,.\, b}\right] \cap A^- = \varnothing}\right\}$

where $A^-$ and $B^-$ denote the closure of $A$ and $B$ in $T$.

Then $A^*$ and $B^*$ are themselves separated sets of $T$.

Proof
From the lemma:


 * $A \subseteq A^*$


 * $B \subseteq B^*$


 * $A^* \cap B^* = \varnothing$

Let $p \notin A^* \cup A^-$.

Thus $p \notin A^*$ and $p \notin A^-$.

Then there exists an open interval $\left({s \,.\,.\, t}\right)$ which is disjoint from $A$ such that $p \in \left({s \,.\,.\, t}\right)$.

Now $\left({s \,.\,.\, t}\right)$ can only intersect $A^*$ only if it intersects some $\left[{a \,.\,.\, b}\right] \subseteq A^*$ where $a, b \in A$.

But we have:
 * $\left({s \,.\,.\, t}\right) \cap A = \varnothing$

and as $a, b \in A$ it follows that:
 * $\left({s \,.\,.\, t}\right) \subseteq \left({a \,.\,.\, b}\right)$

That means $p \in A^*$.

But we have $p \notin A^*$.

Therefore:
 * $\left({s \,.\,.\, t}\right) \cap A^* = \varnothing$

Thus:
 * $p \notin \left({A^*}\right)^-$

Hence:

Hence the result.