Definition:Real Interval

Definition
The set of all real numbers between any two given real numbers $a$ and $b$ is called a (real) interval. There are many kinds of intervals, each more-or-less consistent with this informal definition.

See below for their rigorous definitions and specific names associated to them.

An arbitrary interval is frequently denoted $\mathbb I$, although some sources use just $I$. Others use $\mathbf I$.

Definitions of Interval Types
It is usual to define intervals in terms of inequalities.

These are in the form of a pair of brackets, either round or square, enclosing the two endpoints of the interval separated by two dots.

Whether the bracket at either end is round or square depends on whether the end point is inside or outside the interval, as specified in the following.

Let $a, b \in \R$.

Also see

 * Open Rectangle, a generalization to higher dimensional spaces
 * Half-Open Rectangle, idem


 * Real Number Line is Metric Space


 * Interval Defined by Betweenness