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 $\sigma \subseteq V$, such that

$\mathrm{\forall }\rho \mathrm{\forall }\sigma :\rho \subseteq \sigma \in \mathrm{\Delta }\Rightarrow \rho \in \mathrm{\Delta }.$

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

$\mathrm{\forall }\rho \mathrm{\forall }\sigma :\rho \subseteq \sigma \in \mathrm{\Gamma }\Rightarrow \rho \in \mathrm{\Gamma }.$

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)