# Definition:Interval/Open Interval

## Definition

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

Let $a, b \in S$.

### Open Interval on Ordered Set

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 $b$.

### Open Real Interval

In the context of the real number line $\R$:

The open (real) interval from $a$ to $b$ is defined as:

$\openint a b := \set {x \in \R: a < x < b}$