order topology in nLab

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> order topology in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="noindex,nofollow" /> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } </style> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script type="text/javascript"> <![CDATA[//><!]]> </script> </head> <body> <div id="Container"> <textarea id='content' readonly=' readonly' rows='24' cols='60' > +-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Topology +-- {: .hide} [[!include topology - contents]] =-- =-- =-- # Contents * table of contents {: toc} ## Definition Given a [[linearly ordered set]] $(S, \lt)$, its __order topology__ is the [[topological structure|topology]] on $S$ generated from the [[subbase of a topology|sub-basis]] $\beta \subset P(S)$ $$ \beta \coloneqq \left\{ (a,\infty), (-\infty,a) \right\}_{a \in S} \,. $$ whose elements are the "open half rays" $$ (a, \infty) \coloneqq \left\{ s \in S \,\vert\, a \lt s \right\} \phantom{AAAA} (-\infty, a) \coloneqq \left\{ s \in S \,\vert\, s \lt a \right\} \,. $$ ## Examples * The usual [[Euclidean space]] [[metric topology]] on the [[real numbers]] is equal to the order topology with respect to the canonical ordering of the real numbers. The analogous statement then holds for the [[rational numbers]] $\mathbb{Q} \subset \mathbb{R}$ equipped with their [[subspace topology]]. ## Related concepts * [[specialization topology]], * [[Scott topology]] ## References * Wikipedia, _[Order topology](https://en.wikipedia.org/wiki/Order_topology)_ [[!redirects order topology]] [[!redirects order topologies]] </textarea> </div>  </body> </html>

CINXE.COM

order topology in nLab