Poset topology

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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 on Δ by letting $\Gamma \subseteq \Delta$ 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.