, the poset topology
associated with a partially ordered
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 , such that
Given a simplicial complex Δ as above, we define a (point set) topology
on Δ by letting Γ be closed
if and only if Γ is a simplicial complex:
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.