Interval (mathematics)
Definitions and terminology

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 − ∞. {\displaystyle -\infty.} Similarly, if the supremum does not exist, one says that the corresponding endpoint is + ∞. {\displaystyle +\infty.}

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 interval notation, which is described below.

An open interval does not include any endpoint, and is indicated with parentheses. For example, (0, 1) = { x ∣ 0 < x < 1 } {\displaystyle (0,1)=\{x\mid 0<x<1\}} 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 (0, +∞) consists of real numbers greater than 0, i.e., positive real numbers. The open intervals are thus one of the forms

where a {\displaystyle a} and b {\displaystyle b} are real numbers such that a ≤ b. {\displaystyle a\leq b.} When a = b {\displaystyle a=b} in the first case, the resulting interval is the empty set (a, a) = ∅, {\displaystyle (a,a)=\varnothing,} 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 or equal to 0 and less than or equal to 1. Closed intervals have one of the following forms in which a and b are real numbers such that a ≤ b: {\displaystyle a\leq b\colon }

The closed intervals are those intervals that are closed sets for the usual topology on the real numbers. The empty set and R {\displaystyle \mathbb {R} } are the only intervals that are both open and closed.

A half-open interval has two endpoints and includes only one of them. It is said left-open or right-open depending on whether the excluded endpoint is on the left or on the right. These intervals are denoted by mixing notations for open and closed intervals. For example, (0, 1 ] means greater than 0 and less than or equal to 1, while [ 0, 1) 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 (− ∞, b ] {\displaystyle (-\infty,b]} and {\displaystyle (a,b]} and, 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 | a − b | /2. 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 [ 0, 1) = { x | 0 ≤ x < 1}, 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 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.
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