Editing Interval (mathematics) (section)

==== 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]].