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)