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