Definition:Interval/Ordered Set/Open

Definition
Let $\left({S, \preceq}\right)$ be an ordered set.

Let $a, b \in S$.

The open interval between $a$ and $b$ is the set:


 * $\left({a \,.\,.\, b}\right) := a^\succ \cap b^\prec = \left\{{s \in S: \left({a \prec s}\right) \land \left({s \prec b}\right)}\right\}$

where:
 * $a^\succ$ denotes the strict upper closure of $a$
 * $b^\prec$ denotes the strict lower closure of $b$.

Also defined as
Some sources require that $a \preceq b$ or $a \prec b$.