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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''.
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| 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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