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| In [[mathematics]], the '''poset topology''' associated with a [[partially ordered]] set ''S'' (or [[poset]] for short) is the [[Alexandrov topology]] (open sets are [[upper set]]s) on the poset of [[finite chain]]s of S, ordered by inclusion.
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| Let V be a set of vertices. An [[abstract simplicial complex]] Δ is a set of finite sets of vertices, known as faces <math>\sigma \subseteq V</math>, such that
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| ::<math>\forall \rho, \sigma. \;\ \rho \subseteq \sigma \in \Delta \Rightarrow \rho \in \Delta</math>
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| Given a simplicial complex Δ as above, we define a (point set) [[topology]] on Δ by letting a subset <math>\Gamma \subseteq \Delta</math> be '''closed''' if and only if Γ is a simplicial complex:
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| ::<math>\forall \rho, \sigma. \;\ \rho \subseteq \sigma \in \Gamma \Rightarrow \rho \in \Gamma</math>
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| This is the [[Alexandrov topology]] on the poset of faces of Δ.
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| The '''order complex''' associated with a poset, S, has the underlying set of S as vertices, and the finite chains (i.e. finite totally-ordered subsets) of S as faces. The poset topology associated with a poset S is the Alexandrov topology on the order complex associated with S.
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| ==See also==
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| * [[Topological combinatorics]]
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| ==External links==
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| * [http://arxiv.org/abs/math/0602226 Poset Topology: Tools and Applications] Michelle L. Wachs, lecture notes IAS/Park City Graduate Summer School in Geometric Combinatorics (July 2004)
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| [[Category:General topology]]
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| [[Category:Order theory]]
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| {{topology-stub}}
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Latest revision as of 14:08, 2 December 2014
The title of the author is Jayson. To play lacross is some thing I truly appreciate performing. Credit authorising is how he tends to make money. Alaska is where I've always been living.
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