Editing Interval (mathematics) (section)

== 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>