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All numbers between two given numbers
intervals of real numbers and some generalizations

In mathematics, a (real) interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. An interval can contain neither endpoint, either endpoint, or both endpoints. For example, the set of real numbers consisting of 0, 1, and all numbers in between is an interval, denoted [0, 1] and called the unit interval; the set of all positive real numbers is an interval, denoted (0, ∞); the set of all real numbers is an interval, denoted (−∞, ∞); and any single real number a is an interval, denoted [a, a]. Intervals are ubiquitous in mathematical analysis. For example, they occur implicitly in the epsilon delta definition of continuity; the intermediate value theorem asserts that the image of an interval by a continuous function is an interval; integrals of real functions are defined over an interval; etc. Interval arithmetic consists of computing with intervals instead of real numbers for providing a guaranteed enclosure of the result of a numerical computation, even in the presence of uncertainties of input data and rounding errors. Intervals are likewise defined on an arbitrary totally ordered set, such as integers or rational numbers. The notation of integer intervals is considered in the special section below.

An interval is a subset of the real numbers that contains all real numbers lying between any two numbers of the subset. The endpoints of an interval are its supremum, and its infimum, if they exist as real numbers. If the infimum does not exist, one says often that the corresponding endpoint is Similarly, if the supremum does not exist, one says that the corresponding endpoint is

Intervals are completely determined by their endpoints and whether each endpoint belong to the interval. This is a consequence of the least upper bound property of the real numbers. This characterization is used to specify intervals by mean of , which is described below. An open interval does not include any endpoint, and is indicated with parentheses. For example, is the interval of all real numbers greater than 0 and less than 1. (This interval can also be denoted by ]0, 1[, see below). The open interval consists of real numbers greater than 0, i. e., positive real numbers. The open intervals are thus one of the forms

where and are real numbers such that When in the first case, the resulting interval is the empty set which is a degenerate interval (see below). The open intervals are those intervals that are open sets for the usual topology on the real numbers. A closed interval is an interval that includes all its endpoints and is denoted with square brackets. For example, means greater than 0 and less than or equal to 1, while means greater than or equal to 0 and less than 1. The half open intervals have the form

Every closed interval is a closed set of the real line, but an interval that is a closed set need not be a closed interval. For example, intervals and are also closed sets in the real line. Intervals and are neither an open set nor a closed set. If one allows an endpoint in the closed side to be an infinity (such as , the result will not be an interval, since it is not even a subset of the real numbers. Instead, the result can be seen as an interval in the extended real line, which occurs in measure theory, for example. In summary, a set of the real numbers is an interval, if and only if it is an open interval, a closed interval, or a half open interval. A degenerate interval is any set consisting of a single real number (i. e., an interval of the form ). Some authors include the empty set in this definition. A real interval that is neither empty nor degenerate is said to be proper, and has infinitely many elements. 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 also commonly known as finite intervals. Bounded intervals are bounded sets, in the sense that their diameter (which is equal to the absolute difference between the endpoints) is finite. The diameter may be called the length, width, measure, range, or size of the interval. The size of unbounded intervals is usually defined as +∞, and the size of the empty interval may be defined as 0 (or left undefined). The centre (midpoint) of a bounded interval with endpoints a and b is (a + b)/2, 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 contains no minimum (an element that is smaller than all other elements); right open if it contains no maximum; and open if it contains neither. The interval , for example, is left closed and right open. The empty set and the set of all reals are both open and closed intervals, while the set of non negative reals, is a closed interval that is right open but not left open. The open intervals are open sets of the real line in its standard topology, and form a base of the open sets. An interval is said to be left closed if it has a minimum element or is left unbounded, right closed if it has a maximum or is right unbounded; it is simply closed if it is both left closed and right closed. So, the closed intervals coincide with the closed sets in that topology. The interior of an interval I is the largest open interval that is contained in I; it is also the set of points in I which are not endpoints of I. The closure of I is the smallest closed interval that contains I; which is also the set I augmented with its finite endpoints. For any set X of real numbers, the interval enclosure or interval span of X is the unique interval that contains X, and does not properly contain any other interval that also contains X. An interval I is a subinterval of interval J if I is a subset of J. An interval I is a proper subinterval of J if I is a proper subset of J. However, there is conflicting terminology for the terms segment and interval, which have been employed in the literature in two essentially opposite ways, resulting in ambiguity when these terms are used. The Encyclopedia of Mathematics defines interval (without a qualifier) to exclude both endpoints (i. e., open interval) and segment to include both endpoints (i. e., closed interval), while Rudin's Principles of Mathematical Analysis calls sets of the form [a, b] intervals and sets of the form (a, b) segments throughout. These terms tend to appear in older works; modern texts increasingly favor the term interval (qualified by open, closed, or half open), regardless of whether endpoints are included.

The interval of numbers between a and b, including a and b, is often denoted The two numbers are called the endpoints of the interval. In countries where numbers are written with a decimal comma, a semicolon may be used as a separator to avoid ambiguity.

To indicate that one of the endpoints is to be excluded from the set, the corresponding square bracket can be either replaced with a parenthesis, or reversed. Both notations are described in International standard ISO 31 11. Thus, in set builder notation,

Each interval , , and represents the empty set, whereas denotes the singleton set When a > b, all four notations are usually taken to represent the empty set. Both notations may overlap with other uses of parentheses and brackets in mathematics. For instance, the notation (a, b) 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. That is why Bourbaki introduced the notation ]a, b[ to denote the open interval. The notation [a, b] too is occasionally used for ordered pairs, especially in computer science. Some authors such as Yves Tillé use ]a, b[ to denote the complement of the interval ; namely, the set of all real numbers that are either less than or equal to a, or greater than or equal to b.

In some contexts, an interval may be defined as a subset of the extended real numbers, the set of all real numbers augmented with −∞ and +∞. 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. Even in the context of the ordinary reals, one may use an infinite endpoint to indicate that there is no bound in that direction. For example, is the set of positive real numbers, also written as The context affects some of the above definitions and terminology. For instance, the interval  =  is closed in the realm of ordinary reals, but not in the realm of the extended reals.

When a and b are integers, the notation ⟦a, b⟧, or or or just a b, is sometimes used to indicate the interval of all integers between a and b included. The notation is used in some programming languages; in Pascal, for example, it is used to formally define a subrange type, most frequently used to specify lower and upper bounds of valid indices of an array. Another way to interpret integer intervals are as sets defined by enumeration, using ellipsis notation. An integer interval that has a finite lower or upper endpoint always includes that endpoint. Therefore, the exclusion of endpoints can be explicitly denoted by writing a b − 1 , a + 1 b , or a + 1 b − 1. Alternate bracket notations like or [a b[ are rarely used for integer intervals.

An open finite interval is a 1 dimensional open ball with a center at and a radius of The closed finite interval is the corresponding closed ball, and the interval's two endpoints form a 0 dimensional sphere. Generalized to dimensional Euclidean space, a ball is the set of points whose distance from the center is less than the radius. In the 2 dimensional case, a ball is called a disk. If a half space is taken as a kind of degenerate ball (without a well defined center or radius), a half space can be taken as analogous to a half bounded interval, with its boundary plane as the (degenerate) sphere corresponding to the finite endpoint.

A finite interval is (the interior of) a 1 dimensional hyperrectangle. Generalized to real coordinate space an axis aligned hyperrectangle (or box) is the Cartesian product of finite intervals. For this is a rectangle; for this is a rectangular cuboid (also called a "box"). Allowing for a mix of open, closed, and infinite endpoints, the Cartesian product of any intervals, is sometimes called an dimensional interval. A facet of such an interval is the result of replacing any non degenerate interval factor by a degenerate interval consisting of a finite endpoint of The faces of comprise itself and all faces of its facets. The corners of are the faces that consist of a single point of

Any finite interval can be constructed as the intersection of half bounded intervals (with an empty intersection taken to mean the whole real line), and the intersection of any number of half bounded intervals is a (possibly empty) interval. Generalized to dimensional affine space, an intersection of half spaces (of arbitrary orientation) is (the interior of) a convex polytope, or in the 2 dimensional case a convex polygon.

An open interval is a connected open set of real numbers. Generalized to topological spaces in general, a non empty connected open set is called a domain.

Intervals of complex numbers can be defined as regions of the complex plane, either rectangular or circular.