• Source: Poset topology

      • 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



      σ
      ⊆
      V


      {\displaystyle \sigma \subseteq V}

      , such that




      ∀
      ρ

      ∀
      σ

      :

      ρ
      ⊆
      σ
      ∈
      Δ
      ⇒
      ρ
      ∈
      Δ
      .


      {\displaystyle \forall \rho \,\forall \sigma \!:\ \rho \subseteq \sigma \in \Delta \Rightarrow \rho \in \Delta .}


      Given a simplicial complex Δ as above, we define a (point set) topology on Δ by declaring a subset



      Γ
      ⊆
      Δ


      {\displaystyle \Gamma \subseteq \Delta }

      be closed if and only if Γ is a simplicial complex, i.e.




      ∀
      ρ

      ∀
      σ

      :

      ρ
      ⊆
      σ
      ∈
      Γ
      ⇒
      ρ
      ∈
      Γ
      .


      {\displaystyle \forall \rho \,\forall \sigma \!:\ \rho \subseteq \sigma \in \Gamma \Rightarrow \rho \in \Gamma .}


      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


      Poset Topology: Tools and Applications Michelle L. Wachs, lecture notes IAS/Park City Graduate Summer School in Geometric Combinatorics (July 2004)

    Kata Kunci Pencarian: