interval

An interval is a subset of a poset (typically the real numbers) which contains every element with an ordering between its endpoints.

Formally,
a closed interval $I$ with endpoints $a,b$ is a subset of a poset $(P,\le )$ such that

where $\le$ is the partial order on set $P$.
An open interval $(a,b)$ is defined similarly, with the corresponding strict partial order.

If the endpoints $a,b$ are finite, then the endpoints are the supremum and infimum of the interval, and the interval is bounded. The length of the interval is $(b-a)$.

Intervals are completely determined by their endpoints and whether each endpoint belongs to the interval. This is a consequence of the least-upper-bound property of the real numbers.

With respect to the set of real numbers $\mathbb{R}$ and its standard ordering:

An open interval does not include any endpoint, and is indicated with parentheses:

A closed interval includes its endpoints, and is denoted with square brackets:

A half-open interval has two endpoints and includes only one of them. It is said to be left-open or right-open depending on whether the excluded endpoint is on the left or on the right. The half-open intervals have the form

See also
nested intervals
partition of an interval