# Definition:Interval/Closed Interval

## Definition

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

Let $a, b \in S$.

### Closed Interval on Ordered Set

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

$\closedint a b := a^\succcurlyeq \cap b^\preccurlyeq = \set {s \in S: \paren {a \preccurlyeq s} \land \paren {s \preccurlyeq b} }$

where:

$a^\succcurlyeq$ denotes the upper closure of $a$
$b^\preccurlyeq$ denotes the lower closure of $b$.

### Closed Real Interval

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

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

$\closedint a b = \set {x \in \R: a \le x \le b}$