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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. | |||
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 | |||
::<math>\forall \rho, \sigma. \;\ \rho \subseteq \sigma \in \Delta \Rightarrow \rho \in \Delta</math> | |||
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: | |||
::<math>\forall \rho, \sigma. \;\ \rho \subseteq \sigma \in \Gamma \Rightarrow \rho \in \Gamma</math> | |||
This is the [[Alexandrov topology]] on the poset of faces of Δ. | |||
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. | |||
==See also== | |||
* [[Topological combinatorics]] | |||
==External links== | |||
* [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) | |||
[[Category:General topology]] | |||
[[Category:Order theory]] | |||
{{topology-stub}} | |||
Revision as of 13:39, 5 January 2014
In mathematics, the poset topology associated with a partially ordered set S (or poset for short) is the Alexandrov topology (open sets are upper sets) on the poset of finite chains of S, ordered by inclusion.
Let V be a set of vertices. An abstract simplicial complex Δ is a set of finite sets of vertices, known as faces , such that
Given a simplicial complex Δ as above, we define a (point set) topology on Δ by letting a subset be closed if and only if Γ is a simplicial complex:
This is the Alexandrov topology on the poset of faces of Δ.
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.
See also
External links
- Poset Topology: Tools and Applications Michelle L. Wachs, lecture notes IAS/Park City Graduate Summer School in Geometric Combinatorics (July 2004)