Definition:Interval/Ordered Set/Open

Definition
Let $\struct {S, \preccurlyeq}$ be an ordered set.

Let $a, b \in S$.

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


 * $\openint a b := a^\succ \cap b^\prec = \set {s \in S: \paren {a \prec s} \land \paren {s \prec b} }$

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 $a \preccurlyeq b$ or $a \prec b$.

Also see

 * Definition:Closed Interval
 * Definition:Half-Open Interval:
 * Definition:Left Half-Open Interval
 * Definition:Right Half-Open Interval