[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/extension-topology-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/extension-topology-wikipedia\/","headline":"Extension topology – Wikipedia","name":"Extension topology – Wikipedia","description":"before-content-x4 In topology, a branch of mathematics, an extension topology is a topology placed on the disjoint union of a","datePublished":"2019-09-17","dateModified":"2019-09-17","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/cd810e53c1408c38cc766bc14e7ce26a?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/cd810e53c1408c38cc766bc14e7ce26a?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4b9c13b1b3f77e42aa49a7717f370f5130ede25e","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4b9c13b1b3f77e42aa49a7717f370f5130ede25e","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/extension-topology-wikipedia\/","wordCount":3657,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4In topology, a branch of mathematics, an extension topology is a topology placed on the disjoint union of a topological space and another set.  There are various types of extension topology, described in the sections below.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Table of ContentsExtension topology[edit]Open extension topology[edit]Closed extension topology[edit]Works cited[edit]Extension topology[edit]Let X be a topological space and P a set disjoint from X. Consider in X\u00a0\u222a\u00a0P the topology whose open sets are of the form A\u00a0\u222a\u00a0Q, where A is an open set of X and Q is a subset of P.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4The closed sets of X\u00a0\u222a\u00a0P are of the form B\u00a0\u222a\u00a0Q, where B is a closed set of X and Q is a subset of P.For these reasons this topology is called the extension topology of X plus P, with which one extends to X\u00a0\u222a\u00a0P the open and the closed sets of X.  As subsets of X\u00a0\u222a\u00a0P the subspace topology of X is the original topology of X, while the subspace topology of P is the discrete topology.  As a topological space, X\u00a0\u222a\u00a0P is homeomorphic to the topological sum of X and P, and X is a clopen subset of X\u00a0\u222a\u00a0P.If Y is a topological space and R is a subset of Y, one might ask whether the extension topology of Y \u2013 R plus R is the same as the original topology of Y, and the answer is in general no.Note the similarity of this extension topology construction and the Alexandroff one-point compactification, in which case, having a topological space X which one wishes to compactify by adding a point \u221e in infinity, one considers the closed sets of X\u00a0\u222a\u00a0{\u221e} to be the sets of the form K, where K is a closed compact set of X, or B\u00a0\u222a\u00a0{\u221e}, where B is a closed set of X.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Open extension topology[edit]Let (X,T){displaystyle (X,{mathcal {T}})} be a topological space and P{displaystyle P} a set disjoint from X{displaystyle X}. The open extension topology of T{displaystyle {mathcal {T}}} plus P{displaystyle P} is  T\u2217=T\u222a{X\u222aA:A\u2282P}.{displaystyle {mathcal {T}}^{*}={mathcal {T}}cup {Xcup A:Asubset P}.}Let X\u2217=X\u222aP{displaystyle X^{*}=Xcup P}. Then T\u2217{displaystyle {mathcal {T}}^{*}}is a topology in X\u2217{displaystyle X^{*}}. The subspace topology of X{displaystyle X} is the original topology of X{displaystyle X}, i.e. T\u2217|X=T{displaystyle {mathcal {T}}^{*}|X={mathcal {T}}}, while the subspace topology of P{displaystyle P} is the discrete topology, i.e. T\u2217|P=P(P){displaystyle {mathcal {T}}^{*}|P={mathcal {P}}(P)}.The closed sets in X\u2217{displaystyle X^{*}} are {B\u222aP:X\u2282B\u2227X\u2216B\u2208T}{displaystyle {Bcup P:Xsubset Bland Xsetminus Bin {mathcal {T}}}}. Note that P{displaystyle P} is closed in X\u2217{displaystyle X^{*}} and X{displaystyle X} is open and dense in X\u2217{displaystyle X^{*}}.If Y a topological space and R is a subset of Y, one might ask whether the open extension topology of Y \u2013 R plus R is the same as the original topology of Y, and the answer is in general no.Note that the open extension topology of X\u2217{displaystyle X^{*}} is smaller than the extension topology of X\u2217{displaystyle X^{*}}.Assuming X{displaystyle X} and P{displaystyle P} are not empty to avoid trivialities, here are a few general properties of the open extension topology:For a set Z and a point p in Z, one obtains the excluded point topology construction by considering in Z the discrete topology and applying the open extension topology construction to Z \u2013 {p} plus p.Closed extension topology[edit]Let X be a topological space and P a set disjoint from X. Consider in X\u00a0\u222a\u00a0P the topology whose closed sets are of the form X\u00a0\u222a\u00a0Q, where Q is a subset of P, or B, where B is a closed set of X.For this reason this topology is called the closed extension topology of X plus P, with which one extends to X\u00a0\u222a\u00a0P the closed sets of X.  As subsets of X\u00a0\u222a\u00a0P the subspace topology of X is the original topology of X, while the subspace topology of P is the discrete topology.The open sets of X\u00a0\u222a\u00a0P are of the form Q, where Q is a subset of P, or A\u00a0\u222a\u00a0P, where A is an open set of X.  Note that P is open in X\u00a0\u222a\u00a0P and X is closed in X\u00a0\u222a\u00a0P.If Y is a topological space and R is a subset of Y, one might ask whether the closed extension topology of Y \u2013 R plus R is the same as the original topology of Y, and the answer is in general no.Note that the closed extension topology of X\u00a0\u222a\u00a0P is smaller than the extension topology of X\u00a0\u222a\u00a0P.For a set Z and a point p in Z, one obtains the particular point topology construction by considering in Z the discrete topology and applying the closed extension topology construction to Z \u2013 {p} plus p.Works cited[edit]     (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/extension-topology-wikipedia\/#breadcrumbitem","name":"Extension topology – Wikipedia"}}]}]