Definition:Closed Interval

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

Let $m, n \in S$. Then the closed interval between $m$ and $n$ is denoted and defined as:
 * $\left[{m \, . \, . \, n}\right] = \begin{cases}

\left\{{x \in S: m \preceq x \land x \preceq n}\right\} & : m \preceq n \\ \varnothing & : n \prec m \end{cases}$

This notation is a fairly recent innovation, and was introduced by C. A. R. Hoare and Lyle Ramshaw.

The older notation, which is more frequently seen, is $\left[{m, n}\right]$. However, it can easily be confused with other usages of the same or similar notation, so its use is deprecated.

Integer Interval
When $S$ is the set $\N$ of natural numbers or $\Z$ of integers, then $\left[{m \,. \, . \, n}\right]$ is called an integer interval.

Also see

 * Closed Real Interval, whose definition is compatible with this.