Poset topology

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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 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, 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

External links

  • Michelle L. Wachs, lecture notes IAS/Park City Graduate Summer School in Geometric Combinatorics (July 2004)

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