Editing Interval (mathematics) (section)

==== 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]].