In mathematics, a simplicial complex is a structured set of simplices (for example, points, line segments, triangles, and their n-dimensional counterparts) such that all the faces and intersections of the elements are also included in the set (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. To distinguish a simplicial complex from an abstract simplicial complex, the former is often called a geometric simplicial complex.
Definitions A simplicial complex K {\displaystyle {\mathcal {K}}} is a set of simplices that satisfies the following conditions:
Every face of a simplex from K {\displaystyle {\mathcal {K}}} is also in K {\displaystyle {\mathcal {K}}} . The non-empty intersection of any two simplices σ 1 , σ 2 ∈ K {\displaystyle \sigma _{1},\sigma _{2}\in {\mathcal {K}}} is a face of both σ 1 {\displaystyle \sigma _{1}} and σ 2 {\displaystyle \sigma _{2}} . See also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry. A simplicial k-complex K {\displaystyle {\mathcal {K}}} is a simplicial complex where the largest dimension of any simplex in K {\displaystyle {\mathcal {K}}} equals k. For instance, a simplicial 2-complex must contain at least one triangle, and must not contain any tetrahedra or higher-dimensional simplices. A pure or homogeneous simplicial k-complex K {\displaystyle {\mathcal {K}}} is a simplicial complex where every simplex of dimension less than k is a face of some simplex σ ∈ K {\displaystyle \sigma \in {\mathcal {K}}} of dimension exactly k. Informally, a pure 1-complex "looks" like it's made of a bunch of lines, a 2-complex "looks" like it's made of a bunch of triangles, etc. An example of a non-homogeneous complex is a triangle with a line segment attached to one of its vertices. Pure simplicial complexes can be thought of as triangulations and provide a definition of polytopes. A facet is a maximal simplex, i.e., any simplex in a complex that is not a face of any larger simplex. (Note the difference from a "face" of a simplex). A pure simplicial complex can be thought of as a complex where all facets have the same dimension. For (boundary complexes of) simplicial polytopes this coincides with the meaning from polyhedral combinatorics. Sometimes the term face is used to refer to a simplex of a complex, not to be confused with a face of a simplex. For a simplicial complex embedded in a k-dimensional space, the k-faces are sometimes referred to as its cells. The term cell is sometimes used in a broader sense to denote a set homeomorphic to a simplex, leading to the definition of cell complex. The underlying space, sometimes called the carrier of a simplicial complex, is the union of its simplices. It is usually denoted by | K | {\displaystyle |{\mathcal {K}}|} or ‖ K ‖ {\displaystyle \|{\mathcal {K}}\|} .
Support The relative interiors of all simplices in K {\displaystyle {\mathcal {K}}} form a partition of its underlying space | K | {\displaystyle |{\mathcal {K}}|} : for each point x ∈ | K | {\displaystyle x\in |{\mathcal {K}}|} , there is exactly one simplex in K {\displaystyle {\mathcal {K}}} containing x {\displaystyle x} in its relative interior. This simplex is called the support of x and denoted supp ( x ) {\displaystyle \operatorname {supp} (x)} .
Closure, star, and link
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