Intervals play an important role in the theory of integration, because they are the simplest sets whose "size" or "measure" or "length" is easy to define. The concept of measure can then be extended to more complicated sets, leading to the Borel measure and eventually to the Lebesgue measure.
Intervals are central to interval arithmetic, a general numerical computing technique that automatically provides guaranteed enclosures for arbitrary formulas, even in the presence of uncertainties, mathematical approximations, and arithmetic roundoff.
To indicate that one of the endpoints is to be excluded from the set, many writers substitute a parenthesis for the corresponding square bracket. Thus, in set builder notation,
International standard ISO 31-11 also defines another notation for intervals, which seems to be more commonly taught in European and South American secondary schools. It uses an inwards pointing bracket to indicate inclusion of the endpoint, and outwards-pointing bracket for exclusion:
Both notations may overlap with other uses of parentheses and brackets in mathematics. For instance, the notation is often used to denote an ordered pair in set theory, the coordinates of a point or vector in analytic geometry and linear algebra, or (sometimes) a complex number in algebra. The notation too is occasionally used for ordered pairs, especially in computer science.
Some authors use to denote the complement of the interval ; namely, the set of all real numbers that are either or .
The notations , , , and are ambiguous. For authors who define intervals as subsets of the real numbers, those notations are either meaningless, or equivalent to the open variants. In the latter case, the interval comprising all real numbers is both open and closed, .
An interval is said to be left-bounded or right-bounded if there is some real number that is, respectively, smaller than or larger than all its elements. An interval is said to be bounded if it is both left- and right-bounded; and is said to be unbounded otherwise. Intervals that are bounded at only one end are said to be half-bounded. The empty set is bounded, and the set of all reals is the only interval that is unbounded at both ends.
Bounded intervals are bounded sets, in the sense that their diameter (the absolute difference between the endpoints) is finite. The diameter may be called the length, width, measure, or size of the interval. The size of unboundded intervals is usually defined as , and the size of the empty interval may be defined as 0 or left undefined.
The center of bounded interval with endpoints and is , and its radius is the half-length . These concepts are undefined for empty or unbounded intervals.
An interval is said to be left-open if and only if it has no minimum (an element that is smaller than all other elements); right-open if it has no maximum; and open if it has both properties. The interval , for example, is left-closed and right-open. The empty set and the set of all reals are open intervals, while the set of non-negative reals, for example, is a right-open but not left-open interval. The open intervals coincide with the open sets of the real line in its standard topology.
An interval is said to be left-closed if it has a minimum element, right-closed if it has a maximum, and simply closed if it hs both. These definitions are usually extended to include the empty set and to the (left- or right-) unbounded intervals, so that the closed intervals coincide with closed sets in that topology.
The interior of an interval is the the largest open interval that is contained in ; it is also the set of points in which are not endpoints of . The closure of is the smallest closed interval that contains ; which is also the set augmented with its finite endpoints.
For any set of real numbers, the interval enclosure or interval span of is the unique interval that contains and is not properly contained in any other interval.
In this interpretation, the notations , , , and are all meaningful and distinct. In particular, denotes the set of all ordinary real numbers, while denotes the extended reals.
This choice affects some of the above defintions and terminology. For instance, the interval is closed in the realm of ordinary reals, but not in the realm of the extended reals.
The intersection of any collection of intervals is always an interval. The union of two open intervals (or two closed intervals) is an interval if andonly only if they have a non-empty intersection.
Any element of an interval defines a partition of into three disjoint intervals : respectively, the elements of that are less than , the singleton , and the elements that are greater than . The parts and are both non-empty (and have non-empty interiors) if an only if is in the interior of . This is an interval version of the trichotomy principle.
A dyadic interval is a bounded real interval whose endpoints are and , where and are integers. Depending on the context, either endpoint may or may not be included in the interval.
Dyadic intervals have some nice properties, such as the following:
The dyadic intervals thus have a structure very similar to an infinite binary tree.