In mathematics, the '''poset topology''' associated to a poset (''S'', ≤) 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 <math>\sigma \subseteq V</math>, such that ::<math>\forall \rho \, \forall \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 declaring a subset <math>\Gamma \subseteq \Delta</math> be '''closed''' if and only if Γ is a simplicial complex, i.e. ::<math>\forall \rho \, \forall \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 to a poset (''S'', ≤) has the set ''S'' as vertices, and the finite chains of (''S'', ≤) as faces. The poset topology associated to a poset (''S'', ≤) is then the Alexandrov topology on the order complex associated to (''S'', ≤).
==See also== * Topological combinatorics
==References== * [https://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
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