About: Order topology

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In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If X is a totally ordered set, the order topology on X is generated by the subbase of "open rays" for all a, b in X. Provided X has at least two elements, this is equivalent to saying that the open intervals together with the above rays form a base for the order topology. The open sets in X are the sets that are a union of (possibly infinitely many) such open intervals and rays.

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rdfs:label	Ordnungstopologie (de) Topologie de l'ordre (fr) 순서 위상 (ko) Order topology (en) Topologia porządkowa (pl) Topologia da ordem (pt) Права порядкова топологія (uk) 序拓撲 (zh)
rdfs:comment	Auf einer total geordneten Menge kann man in natürlicher Weise eine Topologie einführen, die mit der Ordnung verträglich ist. Diese Topologie wird Ordnungstopologie genannt. Einige Begriffe aus Topologie und Metrik wie diskret, dicht und vollständig lassen sich so auf Ordnungen übertragen. (de) En mathématiques, la topologie de l'ordre est une topologie naturelle définie sur tout ensemble ordonné (E, ≤), et qui dépend de la relation d'ordre ≤. Lorsque l'on définit la topologie usuelle de la droite numérique ℝ, deux approches équivalentes sont possibles. On peut se fonder sur la relation d'ordre dans ℝ, ou sur la valeur absolue de la distance entre deux nombres. Les égalités ci-dessous permettent de passer de l'une à l'autre : La valeur absolue se généralise en la notion de distance, qui induit le concept de topologie d'un espace métrique. Nous nous intéressons ici à l'autre approche. (fr) 순서론에서 순서 위상(順序位相, 영어: order topology)은 전순서 집합 위의, 열린구간으로부터 생성되는 위상이다. (ko) Topologia porządkowa – topologia wyznaczona przez porządek liniowy w pewnym zbiorze. Naturalnym przykładem topologii porządkowej jest prosta rzeczywista z topologią generowaną przez przedziały otwarte. (pl) A topologia da ordem é a topologia associada a uma relação de ordem em um conjunto. (pt) Пра́ва поря́дкова тополо́гія — топологія на лінійно впорядкованій множині , породжена множинами вигляду = { ∈ \| > }, ∈ . (uk) 数学上，序拓撲是可以定義在任意全序集上的拓扑结构。此為將实数的拓撲結構推廣到任意全序集上所得。具有此種拓撲結構的拓撲空間稱為序空間。如果 X 為全序集，則 X 的序拓扑由無界開區間組成的準基生成，其中 a,b 取遍 X 的所有元素。這等價於，開區间連同上述無界開區間組成序拓撲的一組基，換言之， X 內的開集可寫成該些開區間和無界開區間的（允許無窮）並。若可對一个拓扑空间 X 的元素定義一個全序，使得該全序給出的序拓撲就是 X 自身的拓撲，則称 X 为可序化的。 X 上的序拓撲使 X 成為一個完全正規的豪斯多夫空间。 R, Q, Z, N 上的標準拓撲均為为序拓扑。 (zh) In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If X is a totally ordered set, the order topology on X is generated by the subbase of "open rays" for all a, b in X. Provided X has at least two elements, this is equivalent to saying that the open intervals together with the above rays form a base for the order topology. The open sets in X are the sets that are a union of (possibly infinitely many) such open intervals and rays. (en)
differentFrom	Order topology (functional analysis)
dcterms:subject	Order theory Topological spaces General topology Ordinal numbers
Wikipage page ID	213650 (xsd:integer)
Wikipage revision ID	1123099174 (xsd:integer)
Link from a Wikipage to another Wikipage	Limit ordinal Order theory Compact space Mathematics Lower limit topology Normal space Second-countable space Clopen Compactification (mathematics) Tychonoff plank Successor ordinal Topological spaces Total order Totally ordered set Hausdorff space Linear continuum J. Arthur Seebach Jr. Filter (mathematics) Base (topology) Direct limit Isolated point Interval (mathematics) Counterexamples in Topology General topology Ordinal numbers Discrete topology Zero-dimensional space Discrete space Boolean algebra (structure) Net (mathematics) Order topology (functional analysis) Ordinal number Rational numbers Real numbers Sequence Set-theoretic Long line (topology) Scattered space Union (set theory) Extremally disconnected space List of topologies Limit point Subspace topology First-countable space First uncountable ordinal Real line Subbase Topological space Partially ordered space Totally disconnected space Basis (topology) Finer topology Completely normal space Lynn Arthur Steen
sameAs	Order topology Order topology Order topology Order topology Order topology Order topology Order topology Order topology Order topology Order topology Order topology Order topology Order topology
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