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Triangulation (topology)
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== Simplicial complexes == {{Main|Abstract simplicial complex|Geometric simplicial complex}} === 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. 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]] === Geometric simplices === Let <math>p_0,...p_n</math> be <math>n+1 </math> affinely independent points in <math>\mathbb{R}^n</math>; i.e. the vectors <math>(p_1-p_0), (p_2-p_0),\dots (p_n-p_0)</math> are [[Linear independence|linearly independent]]. The set <math display=inline>\Delta = \{ \sum_{i=0}^n t_ip_i \,|\, \text{each}\, t_i \in [0,1] \,\text{and}\, \sum_{i=0}^n t_i = 1\}</math> is said to be the ''simplex spanned by <math>p_0,...p_n</math>''. It has ''dimension'' <math>n</math> by definition. The points <math>p_0,...p_n</math> are called the vertices of <math> \Delta </math>, the simplices spanned by <math>n</math> of the <math>n+1</math> vertices are called faces, and the boundary <math>\partial \Delta</math> is defined to be the union of the faces. The <math>n</math>''-dimensional standard-simplex'' is the simplex spanned by the [[unit vector]]s <math> e_0,...e_n</math><ref name=":04">{{citation|surname1=James R. Munkres|title=Elements of algebraic topology|volume=1984|publisher=Addison Wesley|publication-place=Menlo Park, California|at=p. 83|isbn=0-201-04586-9|date=1984 }}</ref> === Geometric simplicial complexes === A geometric simplicial complex <math>\mathcal{S}\subseteq\mathcal{P}(\mathbb{R}^n)</math> is a collection of geometric simplices such that * If <math>S</math> is a simplex in <math>\mathcal{S}</math>, then all its faces are in <math>\mathcal{S}</math>. * If <math>S, T</math> are two distinct simplices in <math>\mathcal{S}</math>, their interiors are disjoint. The union of all the simplices in <math>\mathcal{S}</math> gives the set of points of <math>\mathcal{S}</math>, denoted <math display=inline>|\mathcal{S}|=\bigcup_{S \in \mathcal{S}} S.</math> This set <math>|\mathcal{S}|</math> is endowed with a topology by choosing the [[closed set]]s to be <math>\{A \subseteq |\mathcal{S}| \;\mid\; A \cap \Delta </math> ''is closed for all'' <math> \Delta \in \mathcal{S}\}</math>. Note that, in general, this topology is not the same as the subspace topology that <math>|\mathcal{S}|</math> inherits from <math>\mathbb{R}^n</math>. The topologies do coincide in the case that each point in the complex lies only in finitely many simplices.<ref name=":04"/> Each geometric complex can be associated with an abstract complex by choosing as a ground set <math>V</math> the set of vertices that appear in any simplex of <math>\mathcal{S}</math> and as system of subsets the subsets of <math>V</math> which correspond to vertex sets of simplices in <math>\mathcal{S}</math>. A natural question is if vice versa, any abstract simplicial complex corresponds to a geometric complex. In general, the geometric construction as mentioned here is not flexible enough: consider for instance an abstract simplicial complex of infinite dimension. However, the following more abstract construction provides a topological space for any kind of abstract simplicial complex: Let <math>\mathcal{T}</math> be an abstract simplicial complex above a set <math>V</math>. Choose a union of simplices <math>(\Delta_F)_{F \in \mathcal{T}}</math>, but each in <math>\mathbb {R}^N</math> of dimension sufficiently large, such that the geometric simplex <math>\Delta_F</math> is of dimension <math>n</math> if the abstract geometric simplex <math>F</math> has dimension <math>n</math>. If <math>E\subset F</math>, <math>\Delta_E\subset \mathbb{R}^N</math>can be identified with a face of <math>\Delta_F\subset\mathbb{R}^M</math> and the resulting topological space is the [[Quotient space (topology)|gluing]] <math>\Delta_E \cup_{i}\Delta_F.</math> Effectuating the gluing for each inclusion, one ends up with the desired topological space. This space is actually unique up to homeomorphism for each choice of <math>\mathcal{T},</math> so it makes sense to talk about ''the'' geometric realization <math>|\mathcal{T}|</math> of <math>\mathcal{T}.</math> [[File:Star link of vertex.png|thumb|150px|A 2-dimensional geometric simplicial complex with vertex V, link(V), and star(V) are highlighted in red and pink.]] As in the previous construction, by the topology induced by gluing, the closed sets in this space are the subsets that are closed in the [[subspace topology]] of every simplex <math>\Delta_F</math> in the complex. The simplicial complex <math>\mathcal{T_n}</math> which consists of all simplices <math>\mathcal{T}</math> of dimension <math>\leq n</math> is called the <math>n</math>''-th skeleton'' of <math>\mathcal{T}</math>. A natural [[neighbourhood (mathematics)|neighbourhood]] of a vertex <math>v \in V</math> in a simplicial complex <math>\mathcal{S}</math> is considered to be given by the [[star (simplicial complex)|star]] <math>\operatorname{star}(v)=\{ L \in \mathcal{S} \;\mid\; v \in L \}</math> of a simplex, whose boundary is the link <math>\operatorname{link}(v)</math>. === Simplicial maps === The maps considered in this category are simplicial maps: Let <math>\mathcal{K}</math>, <math>\mathcal{L}</math> be abstract simplicial complexes above sets <math>V_K</math>, <math>V_L</math>. A simplicial map is a function <math>f:V_K \rightarrow V_L</math> which maps each simplex in <math>\mathcal{K}</math> onto a simplex in <math>\mathcal{L}</math>. By affine-linear extension on the simplices, <math>f </math> induces a map between the geometric realizations of the complexes.<ref name=":04"/> === Examples === * Let <math>W =\{a,b,c,d,e,f\}</math> and let <math>\mathcal{T} = \Big\{ \{a\}, \{b\},\{c\},\{d\},\{e\},\{f\}, \{a,b\},\{a,c\},\{a,d\},\{a,e\},\{a,f\}\Big\}</math>. The associated geometric complex is a star with center <math>\{a\}</math>. * Let <math>V= \{A,B,C,D\}</math> and let <math>\mathcal{S} = \mathcal{P}(V)</math>. Its geometric realization <math>|\mathcal{S}|</math> is a [[tetrahedron]]. * Let <math>V</math> as above and let <math>\mathcal{S}' =\; \mathcal{P}(V)\setminus \{A,B,C,D\}</math>. The geometric simplicial complex is the [[Boundary (topology)|boundary]] of a tetrahedron <math>|\mathcal{S'}| = \partial |\mathcal{S}|</math>.
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