Poset topology
In mathematics, the poset topology associated to a poset is the Alexandrov topology on the poset of finite chains of, 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 topology on Δ by declaring a subset be closed if and only if Γ is a simplicial complex, i.e.
This is the Alexandrov topology on the poset of faces of Δ.
The order complex associated to a poset has the set S as vertices, and the finite chains of as faces. The poset topology associated to a poset is then the Alexandrov topology on the order complex associated to.
