mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the poset topology associated to a poset (''S'', ≤) is the

Alexandrov topology In topology, an Alexandrov topology is a topology in which the intersection of any family of open sets is open. It is an axiom of topology that the intersection of any ''finite'' family of open sets is open; in Alexandrov topologies the finite rest ...

(open sets are

upper set In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in ''X'') of a partially ordered set (X, \leq) is a subset S \subseteq X with the following property: if ''s'' is in ''S'' and if ''x'' in ''X'' is larger ...

s) on the poset of finite

chains A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. A c ...

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 \, \forall \sigma \!: \ \rho \subseteq \sigma \in \Delta \Rightarrow \rho \in \Delta.

Given a simplicial complex Δ as above, we define a (point set)

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

on Δ by declaring a subset

\Gamma \subseteq \Delta

be closed if and only if Γ is a simplicial complex, i.e. ::

\forall \rho \, \forall \sigma \!: \ \rho \subseteq \sigma \in \Gamma \Rightarrow \rho \in \Gamma.

This is the

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'', ≤).

References

Poset Topology: Tools and Applications

Michelle L. Wachs Michelle Lynn Wachs is an American mathematician who specializes in algebraic combinatorics and works as a professor of mathematics at the University of Miami.

, lecture notes IAS/Park City Graduate Summer School in Geometric Combinatorics (July 2004) General topology Order theory {{topology-stub

See also

References