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 \and 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.