Definition:Interval/Ordered Set/Left Half-Open

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

Let $a, b \in S$.

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


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

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

Also defined as
Some sources require that $a \preccurlyeq b$.

Also see

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