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 [math]\displaystyle{ \sigma \subseteq V }[/math], such that

[math]\displaystyle{ \forall \rho \, \forall \sigma \!: \ \rho \subseteq \sigma \in \Delta \Rightarrow \rho \in \Delta. }[/math]

Given a simplicial complex Δ as above, we define a (point set) topology on Δ by declaring a subset [math]\displaystyle{ \Gamma \subseteq \Delta }[/math] be closed if and only if Γ is a simplicial complex, i.e.

[math]\displaystyle{ \forall \rho \, \forall \sigma \!: \ \rho \subseteq \sigma \in \Gamma \Rightarrow \rho \in \Gamma. }[/math]

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, ≤).

See also

• Topological combinatorics

References

• Poset Topology: Tools and Applications Michelle L. Wachs, lecture notes IAS/Park City Graduate Summer School in Geometric Combinatorics (July 2004)