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| In [[mathematics]], the '''excluded point topology''' is a [[topological space|topology]] where exclusion of a particular point defines [[open set|openness]]. Formally, let ''X'' be any set and ''p'' ∈ ''X''. The collection
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| :''T'' = {''S'' ⊆ ''X'': ''p'' ∉ ''S'' or ''S'' = X;}
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| of subsets of ''X'' is then the excluded point topology on ''X''. | |
| There are a variety of cases which are individually named:
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| * If ''X'' has two points we call it the '''[[Sierpiński space]]'''. This case is somewhat special and is handled separately.
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| * If ''X'' is [[finite set|finite]] (with at least 3 points) we call the topology on ''X'' the '''finite excluded point topology'''
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| * If ''X'' is [[countably infinite]] we call the topology on ''X'' the '''countable excluded point topology'''
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| * If ''X'' is [[uncountable]] we call the topology on ''X'' the '''uncountable excluded point topology'''
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| A generalization / related topology is the [[open extension topology]]. That is if <math>X\backslash \{p\} </math> has the discrete topology then the open extension topology will be the excluded point topology.
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| This topology is used to provide interesting examples and counterexamples. Excluded point topology is also connected and that is clear since the only open set containing the excluded point is X itself and hence X cannot be written as disjoint union of two proper open subsets.
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| ==See also==
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| * [[Sierpiński space]]
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| * [[Particular point topology]]
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| * [[Alexandrov topology]]
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| * [[Finite topological space]]
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| * [[Fort space]]
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| ==References==
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| *{{Citation | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[[Counterexamples in Topology]] | origyear=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446 | year=1995}}
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| my notes Taha el Turki.[1]
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| [[Category:Topological spaces]]
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Hi there, I am Andrew Berryhill. I am really fond of to go to karaoke but I've been using on new issues lately. Her family life in Alaska but her husband desires them to transfer. Distributing manufacturing has been his profession for some time.
Here is my page :: real psychics, inquiry,