Poset topology
Encyclopedia
In mathematics
, the poset topology associated with a partially ordered set S (or poset for short) is the Alexandrov topology
(open sets are upper set
s) 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
, such that
Given a simplicial complex Δ as above, we define a (point set) topology
on Δ by letting a subset
be closed if and only if Γ is a simplicial complex:
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the poset topology associated with a partially ordered set S (or poset for short) is the Alexandrov topology
Alexandrov topology
In topology, an Alexandrov space is a topological space 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...
(open sets are upper set
Upper set
In mathematics, an upper set of a partially ordered set is a subset U with the property that x is in U and x≤y imply y is in U....
s) on the poset of finite chains of S, ordered by inclusion.
Let V be a set of vertices. An abstract simplicial complex
Abstract simplicial complex
In mathematics, an abstract simplicial complex is a purely combinatorial description of the geometric notion of a simplicial complex, consisting of a family of finite sets closed under the operation of taking subsets...
Δ is a set of finite sets of vertices, known as faces
, such that
Given a simplicial complex Δ as above, we define a (point set) topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
on Δ by letting a subset
be closed if and only if Γ is a simplicial complex:
This is the Alexandrov topology
Alexandrov topology
In topology, an Alexandrov space is a topological space 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...
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.
External links
- Poset Topology: Tools and Applications Michelle L. Wachs, lecture notes IAS/Park City Graduate Summer School in Geometric Combinatorics (July 2004)