Definitions

# Poset topology

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 $sigma subseteq V$, such that

$forall rho, sigma. ; rho subseteq sigma in Delta Rightarrow rho in Delta$
Given a simplicial complex Δ as above, we define a (point set) topology $Gamma subseteq Delta$ on Δ by letting Γ be closed if and only if Γ is a simplicial complex:
$forall rho, sigma. ; rho subseteq sigma in Gamma Rightarrow rho in Gamma$
This is the Alexandrov topology on the poset of faces of Δ.

The order complex associated 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.