Editing Extended real number line (section)

==Order and topological properties==

The extended real number system <math>\overline{\R}</math>, defined as <math>[-\infty,+\infty]</math> or <math>\R\cup\left\{-\infty,+\infty\right\}</math>, can be turned into a [[totally ordered set]] by defining <math>-\infty\leq a\leq+\infty</math> for all <math>a\in\overline{\R}</math>. With this [[order topology]], <math>\overline{\R}</math> has the desirable property of [[compact space|compactness]]: Every [[subset]] of <math>\overline\R</math> has a [[supremum]] and an [[infimum]]<ref name=":1">{{cite book |last1=Oden |first1=J. Tinsley |last2= Demkowicz|first2= Leszek|title=Applied Functional Analysis |date=16 January 2018 |publisher=Chapman and Hall/CRC |isbn=9781498761147 |page=74 |edition=3 |access-date=8 December 2019 |url=https://www.crcpress.com/Applied-Functional-Analysis/Oden-Demkowicz/p/book/9781498761147}}</ref> (the infimum of the [[empty set]] is <math>+\infty</math>, and its supremum is <math>-\infty</math>). Moreover, with this [[topological space|topology]], <math>\overline\R</math> is [[homeomorphic]] to the [[unit interval]] <math>[0,1]</math>. Thus the topology is [[metrizable]], corresponding (for a given homeomorphism) to the ordinary [[metric (mathematics)|metric]] on this interval. There is no metric, however, that is an extension of the ordinary metric on <math>\R</math>.

In this topology, a set <math>U</math> is a [[neighborhood (mathematics)|neighborhood]] of <math>+\infty</math> if and only if it contains a set <math>\{x:x>a\}</math> for some real number <math>a</math>. The notion of the neighborhood of <math>-\infty</math> can be defined similarly. Using this characterization of extended-real neighborhoods, [[limit of a function|limits]] with <math>x</math> tending to <math>+\infty</math> or <math>-\infty</math>, and limits "equal" to <math>+\infty</math> and <math>-\infty</math>, reduce to the general topological definition of limits—instead of having a special definition in the real number system.