Editing Interval (mathematics)

{{Short description|All numbers between two given numbers}}
{{About|intervals of real numbers and some generalizations|intervals in order theory|Interval (order theory)|other uses|Interval (disambiguation)}}

[[File:Interval0.png|thumb|400px|The addition ''x'' + ''a'' on the number line. All numbers greater than ''x'' and less than ''x'' + ''a'' fall within that open interval.]]
In [[mathematics]], a  '''real interval''' is the [[set (mathematics)|set]] of all [[real number]]s 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 [[Bounded set|bound]]. A real interval can contain neither endpoint, either endpoint, or both endpoints, excluding any endpoint which is infinite.

For example, the set of real numbers consisting of {{math|0}}, {{math|1}}, and all numbers in between is an interval, denoted {{math|[0, 1]}} and called the [[unit interval]]; the set of all [[positive real numbers]] is an interval, denoted {{math|(0, ∞)}}; the set of all real numbers is an interval, denoted {{math|(−∞, ∞)}}; and any single real number {{mvar|a}} is an interval, denoted {{math|[''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; [[integral]]s of [[real function]]s 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 error]]s.

Intervals are likewise defined on an arbitrary [[total order|totally ordered]] set, such as [[integers]] or [[rational numbers]]. The notation of integer intervals is considered [[#Integer intervals|in the special section below]].

{{hatnote|Unless explicitly otherwise specified, all intervals considered in this article are real intervals, that is, intervals of real numbers. Notable generalizations are summarized in a section below possibly with links to separate articles.}}

==Definitions and terminology==
An ''interval'' is a [[subset]] of the [[real number]]s that contains all real numbers lying between any two numbers of the subset. In particular, the [[empty set]] <math>\varnothing</math> and the entire set of real numbers <math>\R</math> are both intervals.

The ''endpoints'' of an interval are its [[supremum]], and its [[infimum]], if they exist as real numbers.<ref name="bertsekas">{{cite book
 | last = Bertsekas | first = Dimitri P.
 | title = Network Optimization: Continuous and Discrete Methods
 | year = 1998
 | url = https://books.google.com/books?id=qUUxEAAAQBAJ&pg=PA409
 | page = 409
 | publisher = Athena Scientific
 | isbn = 1-886529-02-7
}}</ref> If the infimum does not exist, one says often that the corresponding endpoint is <math>-\infty.</math> Similarly, if the supremum does not exist, one says that the corresponding endpoint is <math>+\infty.</math>

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 ''{{vanchor|interval notation}}'', which is described below.

An '''''{{visible anchor|open interval}}''''' does not include any endpoint, and is indicated with parentheses.<ref name="strichartz">{{cite book
 | last = Strichartz | first = Robert S.
 | title = The Way of Analysis
 | year = 2000
 | url = https://books.google.com/books?id=Yix09oVvI1IC&pg=PA86
 | page = 86
 | publisher = Jones & Bartlett Publishers
 | isbn = 0-7637-1497-6
}}</ref> For example, <math>(0, 1) = \{x \mid 0 < x < 1\}</math> is the interval of all real numbers greater than {{math|0}} and less than {{math|1}}. (This interval can also be denoted by {{math|]0, 1[}}, see below). The open interval {{math|{{open-open|0, +∞}}}} consists of real numbers greater than {{math|0}}, i.e., positive real numbers. The open intervals have thus one of the forms
:<math>\begin{align}
 (a,b)        &= \{x\in\mathbb R \mid a<x<b\}, \\
 (-\infty, b) &= \{x\in\mathbb R \mid x<b\}, \\
 (a, +\infty) &= \{x\in\mathbb R \mid a<x\}, \\
 (-\infty, +\infty) &= \R, \\
 (a,a)&=\emptyset,
\end{align}</math>
where <math>a</math> and <math>b</math> are real numbers such that <math>a< b.</math> In the last case, the resulting interval is the [[empty set]] and does not depend on {{tmath|a}}. The open intervals are those intervals that are [[open set]]s for the usual [[topological space|topology]] on the real numbers.

A '''''{{visible anchor|closed interval}}''''' is an interval that includes all its endpoints and is denoted with square brackets.<ref name="strichartz" /> For example, {{closed-closed|0, 1}} means greater than or equal to {{math|0}} and less than or equal to {{math|1}}. Closed intervals have one of the following forms in which {{mvar|a}} and {{mvar|b}} are real numbers such that <math>a<  b\colon</math>
:<math>\begin{align}
  \;[a,b]        &= \{x\in\mathbb R \mid a\le x\le b\}, \\
 \left(-\infty, b\right] &= \{x\in\mathbb R \mid x\le b\}, \\
 \left[a, +\infty\right) &= \{x\in\mathbb R \mid a\le x\}, \\
 (-\infty, +\infty) &= \R,\\
 \left[a,a\right]&=\{a\}.
\end{align}</math>
The closed intervals are those intervals that are [[closed set]]s for the usual [[topological space|topology]] on the real numbers.

A ''{{visible anchor|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.<ref name=":2">{{Cite web|last=Weisstein|first=Eric W.|title=Interval|url=https://mathworld.wolfram.com/Interval.html|access-date=2020-08-23|website=mathworld.wolfram.com|language=en}}</ref> For example, {{open-closed|0, 1}} means greater than {{math|0}} and less than or equal to {{math|1}}, while {{closed-open|0, 1}} means greater than or equal to {{math|0}} and less than {{math|1}}. The half-open intervals have the form
:<math>\begin{align}
 \left(a,b\right]       &= \{x\in\R \mid a<x\le b\}, \\
 \left[a,b\right)       &= \{x\in\R \mid a\le x<b\}. \\
\end{align}</math>

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. The only intervals that appear twice in the above classification are {{tmath|\emptyset}} and {{tmath|\R}} that are both open and closed.<ref name="eom">{{eom|title=Interval and segment}}</ref><ref name="tao">{{cite book
 | last = Tao | first = Terence | author-link = Terence Tao
 | title = Analysis I
 | year = 2016
 | url = https://books.google.com/books?id=ecTsDAAAQBAJ&pg=PA212
 | page = 212
 | edition = 3
 | series = Texts and Readings in Mathematics
 | volume = 37
 | publisher = Springer
 | location = Singapore
 | isbn = 978-981-10-1789-6
 | issn = 2366-8725
 | doi = 10.1007/978-981-10-1789-6
 | lccn = 2016940817
}} See Definition 9.1.1.</ref>

A ''{{visible anchor|degenerate interval}}'' is any [[singleton set|set consisting of a single real number]] (i.e., an interval of the form {{closed-closed|''a'', ''a''}}).<ref name="cramer">{{cite book
 | last = Cramér | first = Harald
 | title = Mathematical Methods of Statistics
 | year = 1999
 | url = https://books.google.com/books?id=CRTKKaJO0DYC&pg=PA11
 | page = 11
 | publisher = Princeton University Press
| isbn = 0691005478
 }}</ref> 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.

{{anchor|bounded interval|unbounded interval|half-bounded interval|finite interval}}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 set]]s, 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 {{math|+∞}}, and the size of the empty interval may be defined as {{math|0}} (or left undefined).

The ''centre'' ([[midpoint]]) of a bounded interval with endpoints {{mvar|a}} and {{mvar|b}} is {{math|(''a'' + ''b'')/2}}, and its ''radius'' is the half-length {{math|{{mabs|''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 {{math|{{closed-open|0, 1}} {{=}} {{mset|''x'' | 0 ≤ ''x'' &lt; 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 set]]s of the real line in its standard [[point-set topology|topology]], and form a [[base (topology)|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 set]]s in that topology.

The ''interior'' of an interval {{mvar|I}} is the largest open interval that is contained in {{mvar|I}}; it is also the set of points in {{mvar|I}} which are not endpoints of {{mvar|I}}. The ''closure'' of {{mvar|I}} is the smallest closed interval that contains {{mvar|I}}; which is also the set {{mvar|I}} augmented with its finite endpoints.

For any set {{mvar|X}} of real numbers, the ''interval enclosure'' or ''interval span'' of {{mvar|X}} is the unique interval that contains {{mvar|X}}, and does not properly contain any other interval that also contains {{mvar|X}}.

An interval {{mvar|I}} is a ''subinterval'' of interval {{mvar|J}} if {{mvar|I}} is a [[subset]] of {{mvar|J}}. An interval {{mvar|I}} is a ''proper subinterval'' of {{mvar|J}} if {{mvar|I}} is a [[proper subset]] of {{mvar|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''<ref>{{Cite web|url=https://www.encyclopediaofmath.org/index.php/Interval_and_segment|title=Interval and segment - Encyclopedia of Mathematics|website=encyclopediaofmath.org|access-date=2016-11-12|url-status=live|archive-url=https://web.archive.org/web/20141226211146/http://www.encyclopediaofmath.org/index.php/Interval_and_segment|archive-date=2014-12-26}}</ref> 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''<ref>{{Cite book|title=Principles of Mathematical Analysis|url=https://archive.org/details/principlesmathem00rudi_663|url-access=limited|last=Rudin|first=Walter|publisher=McGraw-Hill|year=1976|isbn=0-07-054235-X|location=New York|pages=[https://archive.org/details/principlesmathem00rudi_663/page/n39 31]}}</ref> 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.

==Notations for intervals==
The interval of numbers between {{mvar|a}} and {{mvar|b}}, including {{mvar|a}} and {{mvar|b}}, is often denoted {{closed-closed|''a'', ''b''}}. 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.

===Including or excluding endpoints===
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]],

:<math>\begin{align}
(a,b) = \mathopen{]}a,b\mathclose{[} &= \{x\in\R \mid a<x<b\},    \\[5mu]
[a,b) = \mathopen{[}a,b\mathclose{[} &= \{x\in\R \mid a\le x<b\}, \\[5mu]
(a,b] = \mathopen{]}a,b\mathclose{]} &= \{x\in\R \mid a<x\le b\}, \\[5mu]
[a,b] = \mathopen{[}a,b\mathclose{]} &= \{x\in\R \mid a\le x\le b\}.
\end{align}</math>

Each interval {{open-open|''a'', ''a''}}, {{closed-open|''a'', ''a''}}, and {{open-closed|''a'', ''a''}} represents the [[empty set]], whereas {{closed-closed|''a'', ''a''}} denotes the singleton set&nbsp;{{math|{''a''}{{null}}}}.  When {{math|''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 {{math|(''a'', ''b'')}} is often used to denote an [[tuple|ordered pair]] in set theory, the [[coordinates]] of a [[point (geometry)|point]] or [[vector (mathematics)|vector]] in [[analytic geometry]] and [[linear algebra]], or (sometimes) a [[complex number]] in [[algebra]]. That is why [[Nicolas Bourbaki|Bourbaki]] introduced the notation {{math|]''a'', ''b''[}} to denote the open interval.<ref>{{cite web|url=http://hsm.stackexchange.com/a/193|title=Why is American and French notation different for open intervals (''x'', ''y'') vs. ]''x'', ''y''[?|website=hsm.stackexchange.com|access-date=28 April 2018}}</ref> The notation {{math|[''a'', ''b'']}} too is occasionally used for ordered pairs, especially in [[computer science]].

Some authors such as Yves Tillé use {{math|]''a'', ''b''[}} to denote the complement of the interval&nbsp;{{open-open|''a'', ''b''}}; namely, the set of all real numbers that are either less than or equal to {{mvar|a}}, or greater than or equal to {{mvar|b}}.

===Infinite endpoints===
In some contexts, an interval may be defined as a subset of the [[extended real number line|extended real numbers]], the set of all real numbers augmented with {{math|−∞}} and {{math|+∞}}.

In this interpretation, the notations {{closed-closed|−∞, ''b''}} , {{open-closed|−∞, ''b''}} , {{closed-closed|''a'', +∞}} , and {{closed-open|''a'', +∞}} are all meaningful and distinct. In particular, {{open-open|−∞, +∞}} denotes the set of all ordinary real numbers, while {{closed-closed|−∞, +∞}} denotes the extended reals. 

Even in the context of the ordinary reals, one may use an [[infinity (mathematics)|infinite]] endpoint to indicate that there is no bound in that direction. For example, {{open-open|0, +∞}} is the set of [[positive real numbers]], also written as <math>\mathbb{R}_+.</math> The context affects some of the above definitions and terminology. For instance, the interval {{open-open|−∞, +∞}}&nbsp;=&nbsp;<math>\R</math> is closed in the realm of ordinary reals, but not in the realm of the extended reals.

===Integer intervals===
When {{mvar|a}} and {{mvar|b}} are [[integer]]s, the notation ⟦''a, b''⟧, or {{closed-closed|''a'' .. ''b''}} or {{math|{''a'' .. ''b''}{{null}}}} or just {{math|''a'' .. ''b''}}, is sometimes used to indicate the interval of all ''integers'' between {{mvar|a}} and {{mvar|b}} included. The notation {{closed-closed|''a'' .. ''b''}} is used in some [[programming language]]s; in [[Pascal programming language|Pascal]], for example, it is used to formally define a subrange type, most frequently used to specify lower and upper bounds of valid [[Indexed family|indices]] of an [[Array data type|array]].

Another way to interpret integer intervals are as [[Set-builder_notation#Sets_defined_by_enumeration|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 {{math|''a'' .. ''b'' − 1}} ,  {{math|''a'' + 1 .. ''b''}} , or  {{math|''a'' + 1 .. ''b'' − 1}}.  Alternate-bracket notations like {{closed-open|''a'' .. ''b''}} or {{math|[''a'' .. ''b''[}} are rarely used for integer intervals.{{citation needed|date=February 2014}}

== Properties ==
The intervals are precisely the [[connected space|connected]] subsets of <math>\R.</math> It follows that the image of an interval by any [[continuous function]] from <math>\mathbb R</math> to <math>\mathbb R</math> is also an interval. This is one formulation of the [[intermediate value theorem]].

The intervals are also the [[convex set|convex subset]]s of <math>\R.</math> The interval enclosure of a subset <math>X\subseteq \R</math> is also the [[convex hull]] of <math>X.</math>

The [[closure (topology)|closure]] of an interval is the union of the interval and the set of its finite endpoints, and hence is also an interval. (The latter also follows from the fact that the closure of every [[connected space|connected subset]] of a [[topological space]] is a connected subset.) In other words, we have{{sfnp|Tao|2016|p=214|loc = See Lemma 9.1.12}}
:<math>\operatorname{cl}(a,b)=\operatorname{cl}(a,b]=\operatorname{cl}[a,b)=\operatorname{cl}[a,b]=[a,b],</math>
:<math>\operatorname{cl}(a,+\infty)=\operatorname{cl}[a,+\infty)=[a,+\infty),</math>
:<math>\operatorname{cl}(-\infty,a)=\operatorname{cl}(-\infty,a]=(-\infty,a],</math>
:<math>\operatorname{cl}(-\infty,+\infty)=(-\infty,\infty).</math>

The intersection of any collection of intervals is always an interval. The union of two intervals is an interval if and only if they have a non-empty intersection or an open end-point of one interval is a closed end-point of the other, for example <math>(a,b) \cup [b,c] = (a,c].</math>

If <math>\R</math> is viewed as a [[metric space]], its [[open ball]]s are the open bounded intervals&nbsp;{{open-open|''c'' + ''r'', ''c'' − ''r''}}, and its [[closed ball]]s are the closed bounded intervals&nbsp;{{closed-closed|''c'' + ''r'', ''c'' − ''r''}}. In particular, the [[metric topology|metric]] and [[order topology|order]] topologies in the real line coincide, which is the standard topology of the real line.

Any element&nbsp;{{mvar|x}} of an interval&nbsp;{{mvar|I}} defines a partition of&nbsp;{{mvar|I}} into three disjoint intervals {{mvar|I}}<sub>1</sub>, {{mvar|I}}<sub>2</sub>, {{mvar|I}}<sub>3</sub>: respectively, the elements of&nbsp;{{mvar|I}} that are less than&nbsp;{{mvar|x}}, the singleton&nbsp;<math>[x,x] = \{x\},</math> and the elements that are greater than&nbsp;{{mvar|x}}. The parts {{mvar|I}}<sub>1</sub> and {{mvar|I}}<sub>3</sub> are both non-empty (and have non-empty interiors), if and only if {{mvar|x}} is in the interior of&nbsp;{{mvar|I}}.  This is an interval version of the [[trichotomy (mathematics)|trichotomy principle]].

==Dyadic intervals==

A ''dyadic interval'' is a bounded real interval whose endpoints are <math>\tfrac{j}{2^n}</math> and <math>\tfrac{j+1}{2^n},</math> where <math>j</math> and <math>n</math> are integers. Depending on the context, either endpoint may or may not be included in the interval.

Dyadic intervals have the following properties:

* The length of a dyadic interval is always an integer power of two.
* Each dyadic interval is contained in exactly one dyadic interval of twice the length.
* Each dyadic interval is spanned by two dyadic intervals of half the length.
* If two open dyadic intervals overlap, then one of them is a subset of the other.

The dyadic intervals consequently have a structure that reflects that of an infinite [[binary tree]].

Dyadic intervals are relevant to several areas of numerical analysis, including [[adaptive mesh refinement]], [[multigrid methods]] and [[wavelet|wavelet analysis]]. Another way to represent such a structure is [[p-adic analysis]] (for {{math|1=''p'' = 2}}).<ref>{{cite journal |last1=Kozyrev |first1=Sergey |year=2002 |title=Wavelet theory as {{mvar|p}}-adic spectral analysis |journal=[[Izvestiya: Mathematics|Izvestiya RAN. Ser. Mat.]] |volume=66 |issue=2 |pages=149–158 |doi=10.1070/IM2002v066n02ABEH000381 |url=http://mi.mathnet.ru/eng/izv/v66/i2/p149 |access-date=2012-04-05|arxiv=math-ph/0012019 |bibcode=2002IzMat..66..367K |s2cid=16796699 }}</ref>

== Generalizations ==

=== Balls ===

An open finite interval <math>(a, b)</math> is a 1-dimensional open [[ball (mathematics)|ball]] with a [[center (geometry)|center]] at <math>\tfrac12(a + b)</math> and a [[radius]] of <math>\tfrac12(b - a).</math> The closed finite interval <math>[a, b]</math> is the corresponding closed ball, and the interval's two endpoints <math>\{a, b\}</math> form a 0-dimensional [[n-sphere|sphere]]. Generalized to <math>n</math>-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 (mathematics)|disk]].

If a [[half-space (geometry)|half-space]] is taken as a kind of [[degeneracy (mathematics)|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.

=== Multi-dimensional intervals ===

A finite interval is (the interior of) a 1-dimensional [[hyperrectangle]]. Generalized to [[real coordinate space]] <math>\R^n,</math> an [[axis-aligned object|axis-aligned]] hyperrectangle (or box) is the [[Cartesian product]] of <math>n</math> finite intervals. For <math>n=2</math> this is a [[rectangle]]; for <math>n=3</math> this is a [[rectangular cuboid]] (also called a "[[box (geometry)|box]]").

Allowing for a mix of open, closed, and infinite endpoints, the Cartesian product of any <math>n</math> intervals, <math>I = I_1\times I_2 \times \cdots \times I_n</math> is sometimes called an '''<math>n</math>-dimensional interval'''.{{cn|date=September 2023}}

A '''facet''' of such an interval <math>I</math> is the result of replacing any non-degenerate interval factor <math>I_k</math> by a degenerate interval consisting of a finite endpoint of <math>I_k.</math> The '''faces''' of <math>I</math> comprise <math>I</math> itself and all faces of its facets. The '''corners''' of <math>I</math> are the faces that consist of a single point of <math>\R^n.</math>{{cn|date=September 2023}}

=== Convex polytopes ===

Any finite interval can be constructed as the [[intersection (set theory)|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 <math>n</math>-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]].

=== Domains ===

An open interval is a connected open set of real numbers. Generalized to [[topological space]]s in general, a non-empty connected open set is called a [[domain (mathematical analysis)|domain]].

===Complex intervals===
Intervals of [[complex number]]s can be defined as regions of the [[complex plane]], either [[rectangle|rectangular]] or [[disk (mathematics)|circular]].<ref>[https://books.google.com/books?id=Vtqk6WgttzcC Complex interval arithmetic and its applications], Miodrag Petković, Ljiljana Petković, Wiley-VCH, 1998, {{ISBN|978-3-527-40134-5}}</ref>

=== Intervals in posets and preordered sets ===
{{main article|interval (order theory)}}

==== Definitions ====
The concept of intervals can be defined in arbitrary [[partially ordered set]]s or more generally, in arbitrary [[preordered set]]s. For a [[preordered set]] <math>(X,\lesssim)</math> and two elements <math>a,b\in X,</math> one similarly defines the intervals<ref name="Vind">{{cite book
|last=Vind
|first=Karl
|title=Independence, additivity, uncertainty
|language=en
|series=Studies in Economic Theory
|volume=14
|publisher=Springer
|location=Berlin
|date=2003
|isbn=978-3-540-41683-8
|doi=10.1007/978-3-540-24757-9
|zbl=1080.91001
}}</ref>{{rp|11, Definition 11}}
:<math>(a,b)            =\{x\in X \mid a<x<b\},</math>
:<math>[a,b]            =\{x\in X \mid a\lesssim x\lesssim b\},</math>
:<math>(a,b]            =\{x\in X \mid a<x\lesssim b\},</math>
:<math>[a,b)            =\{x\in X \mid a\lesssim x<b\},</math>
:<math>(a,\infty)       =\{x\in X \mid a<x\},</math>
:<math>[a,\infty)       =\{x\in X \mid a\lesssim x\},</math>
:<math>(-\infty,b)      =\{x\in X \mid x<b\},</math>
:<math>(-\infty,b]      =\{x\in X \mid x\lesssim b\},</math>
:<math>(-\infty,\infty) =X,</math>
where <math>x<y</math> means <math>x\lesssim y\not\lesssim x.</math> Actually, the intervals with single or no endpoints are the same as the intervals with two endpoints in the larger preordered set
:<math>\bar X=X\sqcup\{-\infty,\infty\}</math>
:<math>-\infty<x<\infty\qquad(\forall x\in X)</math>
defined by adding new smallest and greatest elements (even if there were ones), which are subsets of <math>X.</math> In the case of <math>X=\mathbb R</math> one may take <math>\bar\mathbb R</math> to be the [[extended real line]].

==== Convex sets and convex components in order theory ====
{{main article|convex set (order theory)}}
A subset <math>A\subseteq X</math> of the [[preordered set]] <math>(X,\lesssim)</math> is '''(order-)convex''' if for every <math>x,y\in A</math> and every <math>x\lesssim z\lesssim y</math> we have <math>z\in A.</math> Unlike in the case of the real line, a convex set of a preordered set need not be an interval. For example, in the [[totally ordered set]] <math>(\mathbb Q,\le)</math> of [[rational number]]s, the set
:<math>\mathbb Q=\{x\in\mathbb Q \mid x^2<2\}</math>
is convex, but not an interval of <math>\mathbb Q,</math> since there is no square root of two in <math>\mathbb Q.</math>

Let <math>(X,\lesssim)</math> be a [[preordered set]] and let <math>Y\subseteq X.</math> The convex sets of <math>X</math> contained in <math>Y</math> form a [[poset]] under inclusion. A [[maximal element]] of this poset is called a '''convex component''' of <math>Y.</math><ref name="Heath">{{cite journal
|last1=Heath
|first1=R. W.
|last2=Lutzer
|first2=David J.
|last3=Zenor
|first3=P. L.
|title=Monotonically normal spaces
|language=en
|journal=Transactions of the American Mathematical Society
|volume=178
|pages=481–493
|date=1973
|issn=0002-9947
|doi=10.2307/1996713
|jstor=1996713
|mr=0372826
|zbl=0269.54009
|doi-access=free
}}</ref>{{rp|Definition 5.1}}<ref name="Steen">{{cite journal
|last=Steen
|first=Lynn A.
|title=A direct proof that a linearly ordered space is hereditarily collection-wise normal
|language=en
|journal=Proceedings of the American Mathematical Society
|volume=24
|pages=727–728
|date=1970
|issue=4
|issn=0002-9939
|doi=10.2307/2037311
|jstor=2037311
|mr=0257985
|zbl=0189.53103
|doi-access=free
}}</ref>{{rp|727}} By the [[Zorn lemma]], any convex set of <math>X</math> contained in <math>Y</math> is contained in some convex component of <math>Y,</math> but such components need not be unique. In a [[totally ordered set]], such a component is always unique. That is, the convex components of a subset of a totally ordered set form a [[partition of a set|partition]].

==== Properties ====
A generalization of the characterizations of the real intervals follows. For a non-empty subset <math>I</math> of a [[linear continuum]] <math>(L,\le),</math> the following conditions are equivalent.<ref name="Munkres">{{cite book
|url=http://www.pearsonhighered.com/bookseller/product/Topology/9780131816299.page
|first=James R.
|last=Munkres
|author-link=James Munkres
|title=Topology
|language=en
|edition=2
|publisher=Prentice Hall
|year=2000
|isbn=978-0-13-181629-9
|zbl=0951.54001
|mr=0464128
}}</ref>{{rp|153, Theorem 24.1}}
* The set <math>I</math> is an interval.
* The set <math>I</math> is order-convex.
* The set <math>I</math> is a connected subset when <math>L</math> is endowed with the [[order topology]].

For a [[subset]] <math>S</math> of a [[lattice (order theory)|lattice]] <math>L,</math> the following conditions are equivalent.
* The set <math>S</math> is a [[sublattice]] and an (order-)convex set.
* There is an [[ideal (order theory)|ideal]] <math>I\subseteq L</math> and a [[filter (mathematics)|filter]] <math>F\subseteq L</math> such that <math>S=I\cap F.</math>

== Applications ==
=== In general topology ===
Every [[Tychonoff space]] is embeddable into a [[product space]] of the closed unit intervals <math>[0,1].</math> Actually, every Tychonoff space that has a [[base (topology)|base]] of [[cardinality]] <math>\kappa</math> is embeddable into the product <math>[0,1]^\kappa</math> of <math>\kappa</math> copies of the intervals.<ref name="Engelking">{{cite book
|first=Ryszard
|last=Engelking
|title=General topology
|language=en
|edition=Revised and completed
|series=Sigma Series in Pure Mathematics
|volume=6
|publisher=Heldermann Verlag
|location=Berlin
|date=1989
|isbn=3-88538-006-4
|mr=1039321
|zbl=0684.54001
}}</ref>{{rp|p. 83, Theorem 2.3.23}}

The concepts of convex sets and convex components are used in a proof that every [[totally ordered set]] endowed with the [[order topology]] is [[completely normal]]<ref name="Steen" /> or moreover, [[monotonically normal]].<ref name="Heath" />

==Topological algebra==
{{more citations needed|section|date=September 2023}}

Intervals can be associated with points of the plane, and hence regions of intervals can be associated with [[region (mathematical analysis)|region]]s of the plane. Generally, an interval in mathematics corresponds to an ordered pair {{math|(''x'', ''y'')}} taken from the [[direct product]] <math>\R \times \R</math> of real numbers with itself, where it is often assumed that {{math|''y'' > ''x''}}. For purposes of [[mathematical structure]], this restriction is discarded,<ref>Kaj Madsen (1979) [https://www.ams.org/mathscinet/pdf/586220.pdf Review of "Interval analysis in the extended interval space" by Edgar Kaucher]{{dead link|date=November 2017 |bot=InternetArchiveBot |fix-attempted=yes }} from [[Mathematical Reviews]]</ref> and "reversed intervals" where {{math|''y'' &minus; ''x'' < 0}} are allowed. Then, the collection of all intervals {{math|[''x'', ''y'']}} can be identified with the [[topological ring]] formed by the [[direct sum of modules#Direct sum of algebras|direct sum]] of <math>\R</math> with itself, where addition and multiplication are defined component-wise.

The direct sum algebra <math>( \R \oplus \R, +, \times)</math> has two [[ideal (ring theory)|ideal]]s, { [''x'',0] : ''x'' ∈ R } and { [0,''y''] : ''y'' ∈ R }. The [[identity element]] of this algebra is the condensed interval {{math|[1, 1]}}. If interval {{math|[''x'', ''y'']}} is not in one of the ideals, then it has [[multiplicative inverse]] {{math|[1/''x'', 1/''y'']}}. Endowed with the usual [[topology]], the algebra of intervals forms a [[topological ring]]. The [[group of units]] of this ring consists of four [[quadrant (plane geometry)|quadrant]]s determined by the axes, or ideals in this case. The [[identity component]] of this group is quadrant I.

Every interval can be considered a symmetric interval around its [[midpoint]].  In a reconfiguration published in 1956 by M Warmus, the axis of "balanced intervals" {{math|[''x'', &minus;''x'']}} is used along with the axis of intervals {{math|[''x'', ''x'']}} that reduce to a point. Instead of the direct sum <math>R \oplus R,</math> the ring of intervals has been identified<ref>[[D. H. Lehmer]] (1956) [https://www.ams.org/mathscinet/pdf/81372.pdf Review of "Calculus of Approximations"]{{dead link|date=November 2017 |bot=InternetArchiveBot |fix-attempted=yes }} from Mathematical Reviews</ref> with the [[hyperbolic number]]s by M. Warmus and [[D. H. Lehmer]] through the identification
:<math>z = \tfrac12(x + y) + \tfrac12(x - y)j,</math>
where <math>j^2 = 1.</math>

This linear mapping of the plane, which amounts of a [[ring isomorphism]], provides the plane with a multiplicative structure having some analogies to ordinary complex arithmetic, such as [[polar decomposition#Alternative planar decompositions|polar decomposition]].

==See also==
*[[Arc (geometry)]]
*[[Inequality (mathematics)|Inequality]]
*[[Interval graph]]
*[[Interval finite element]]
*[[Interval (statistics)]]
*[[Line segment]]
*[[Partition of an interval]]
*[[Unit interval]]

==References==
{{reflist}}

== Bibliography ==
* T. Sunaga, [http://www.cs.utep.edu/interval-comp/sunaga.pdf "Theory of interval algebra and its application to numerical analysis"] {{Webarchive|url=https://web.archive.org/web/20120309164347/http://www.cs.utep.edu/interval-comp/sunaga.pdf |date=2012-03-09 }}, In: Research Association of Applied Geometry (RAAG) Memoirs, Ggujutsu Bunken Fukuy-kai. Tokyo, Japan, 1958, Vol. 2, pp.&nbsp;29–46 (547-564); reprinted in Japan Journal on Industrial and Applied Mathematics, 2009, Vol. 26, No. 2-3, pp.&nbsp;126–143.

==External links==
* ''A Lucid Interval'' by Brian Hayes: An [http://www.americanscientist.org/issues/pub/a-lucid-interval American Scientist article] provides an introduction.
*[http://www.cs.utep.edu/interval-comp/main.html Interval computations website] {{Webarchive|url=https://web.archive.org/web/20060302095039/http://www.cs.utep.edu/interval-comp/main.html |date=2006-03-02 }}
*[http://www.cs.utep.edu/interval-comp/icompwww.html Interval computations research centers] {{Webarchive|url=https://web.archive.org/web/20070203144604/http://www.cs.utep.edu/interval-comp/icompwww.html |date=2007-02-03 }}
* [http://demonstrations.wolfram.com/IntervalNotation/ Interval Notation] by George Beck, [[Wolfram Demonstrations Project]].
* {{MathWorld |title=Interval |urlname=Interval}}

{{DEFAULTSORT:Interval (Mathematics)}}
[[Category:Sets of real numbers]]
[[Category:Order theory]]
[[Category:Topology]]