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 ${\displaystyle \sigma \subseteq V}$, such that

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

Given a simplicial complex Δ as above, we define a (point set) topology on Δ by declaring a subset ${\displaystyle \Gamma \subseteq \Delta }$ be closed if and only if Γ is a simplicial complex, i.e.

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

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, ≤).