Editing Triangulation (topology) (section)

=== Abstract simplicial complexes ===
An abstract simplicial complex above a set <math>V</math> is a system <math>\mathcal{T} \subset \mathcal{P}(V)</math> of non-empty subsets such that:

* <math>\{v_0\} \in \mathcal{T}</math> for each <math>v_0\in V</math>;
* if <math>E \in \mathcal{T}</math> and <math>\emptyset \neq F\subset E,</math> then <math>F \in \mathcal{T}</math>.

The elements of <math>\mathcal{T}</math> are called ''simplices,'' the elements of <math>V</math> are called ''vertices.'' A simplex with <math>n+1</math> vertices has ''dimension'' <math>n</math> by definition. The dimension of an abstract simplicial complex is defined as <math>\text{dim}(\mathcal{T})= \text{sup}\;\{\text{dim}(F):F \in \mathcal{T}\} \in \mathbb{N}\cup \infty</math>.<ref name=":12">{{citation|surname1=John M. Lee|editor-surname1= Springer Verlag|title=Introduction to Topological manifolds|publisher=Springer Verlag|publication-place=New York/Berlin/Heidelberg|at=p.&nbsp;92|isbn=0-387-98759-2|date=2000
}}</ref>

Abstract simplicial complexes can be realized as geometrical objects by associating each abstract simplex with a geometric simplex, defined below.
[[File:Geometric simplices in dimension 1,2 and 3.png|thumb|200px|Geometric simplices in dimension 1, 2 and 3]]