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| In mathematics, more specifically [[general topology]], the '''double origin topology''' is an example of a [[topology]] given to the plane '''R'''<sup>2</sup> with an extra point, say 0*, added. In this case, the double origin topology gives a topology on the set {{nowrap|1=''X'' = '''R'''<sup>2</sup> ∐ {0*} }}, where ∐ denotes the [[disjoint union]].
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| == Construction ==
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| Given a point ''x'' belonging to ''X'', such that {{nowrap|1=''x'' ≠ 0}} and {{nowrap|1=''x'' ≠ 0*}}, the [[Neighbourhood (mathematics)|neighbourhood]]s of ''x'' are those given by the standard [[Metric_topology#Open_and_closed_sets.2C_topology_and_convergence|metric topology]] on {{nowrap|1='''R'''<sup>2</sup>−{0}.}}<ref name="CEIT2">{{Citation|first=L. A.|last=Steen|first2=J. A.|last2=Seebach|title=[[Counterexamples in Topology]]|pages=92 − 93|publisher=Dover|year=1995|ISBN=0-486-68735-X}}</ref> We define a [[countably infinite]] [[Base (topology)|basis]] of neighbourhoods about the point 0 and about the additional point 0*. For the point 0, the basis, [[Indexed family|indexed]] by ''n'', is defined to be:<ref name="CEIT2"/>
| | НАША КОМАНДА<br><br>КОНСТАНТИН<br>дизайнер <br><br>Я был на Вашем месте, я тоже заказывал дизайн. Обращайтесь и я Вас удивлю!<br><br>ЕКАТЕРИНА <br>режиссер <br><br>Я арт-менеджер и мое любимое дело - это фото-режиссура.<br><br><br>РИЧАРД<br>кот <br><br>Я символ студии и я слежу за работой Кати и Кости.<br><br>http://www.kks.by<br><br>Feel free to visit my blog ... [http://kks.by/?cat=29 Design & Fashion Studio "KK"] |
| :<math> \ N(0,n) = \{ (x,y) \in {\bold R}^2 : x^2 + y^2 < 1/n^2, \ y > 0\} \cup \{0\} . </math>
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| In a similar way, the basis of neighbourhoods of 0* is defined to be:<ref name="CEIT2"/>
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| :<math>N(0^*,n) = \{ (x,y) \in {\bold R}^2 : x^2 + y^2 < 1/n^2, \ y < 0\} \cup \{0^*\} . </math>
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| == Properties ==
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| The space {{nowrap|1='''R'''<sup>2</sup> ∐ {0*}}}, along with the double origin topology is an example of a [[Hausdorff space]], although it is not [[completely Hausdorff]]. In terms of compactness, the space {{nowrap|1='''R'''<sup>2</sup> ∐ {0*}}}, along with the double origin topology fails to be either [[compact (topology)|compact]], [[paracompact]] or [[locally compact]], however, ''X'' is [[second countable]]. Finally, it is an example of an [[Connected_space#Arc_connectedness|arc connected]] space.<ref name="CEIT3">{{Citation|first=L. A.|last=Steen|first2=J. A.|last2=Seebach|title=[[Counterexamples in Topology]]|pages=198 – 199|publisher=Dover|year=1995|ISBN=0-486-68735-X}}</ref>
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| == References == | |
| {{reflist}}
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| {{DEFAULTSORT:Double Origin Topology}}
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| [[Category:General topology]]
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НАША КОМАНДА
КОНСТАНТИН
дизайнер
Я был на Вашем месте, я тоже заказывал дизайн. Обращайтесь и я Вас удивлю!
ЕКАТЕРИНА
режиссер
Я арт-менеджер и мое любимое дело - это фото-режиссура.
РИЧАРД
кот
Я символ студии и я слежу за работой Кати и Кости.
http://www.kks.by
Feel free to visit my blog ... Design & Fashion Studio "KK"