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  2. Abstract simplicial complex - Wikipedia

    en.wikipedia.org/wiki/Abstract_simplicial_complex

    The finite sets that belong to Δ are called faces of the complex, and a face Y is said to belong to another face X if Y ⊆ X, so the definition of an abstract simplicial complex can be restated as saying that every face of a face of a complex Δ is itself a face of Δ. The vertex set of Δ is defined as V(Δ) = ∪Δ, the union of all faces ...

  3. Simplicial complex - Wikipedia

    en.wikipedia.org/wiki/Simplicial_complex

    A simplicial 3-complex. In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional counterparts (see illustration). ). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy th

  4. Čech complex - Wikipedia

    en.wikipedia.org/wiki/Čech_complex

    Constructing the Čech complex of a set of points sampled from a circle. In algebraic topology and topological data analysis, the Čech complex is an abstract simplicial complex constructed from a point cloud in any metric space which is meant to capture topological information about the point cloud or the distribution it is drawn from.

  5. Clique complex - Wikipedia

    en.wikipedia.org/wiki/Clique_complex

    The clique complex X(G) of an undirected graph G is an abstract simplicial complex (that is, a family of finite sets closed under the operation of taking subsets), formed by the sets of vertices in the cliques of G.

  6. Simplicial homology - Wikipedia

    en.wikipedia.org/wiki/Simplicial_homology

    Simplicial homology is defined by a simple recipe for any abstract simplicial complex. It is a remarkable fact that simplicial homology only depends on the associated topological space. [2]: sec.8.6 As a result, it gives a computable way to distinguish one space from another.

  7. Subdivision (simplicial complex) - Wikipedia

    en.wikipedia.org/wiki/Subdivision_(simplicial...

    Let K be an abstract simplicial complex (ASC). The face poset of K is a poset made of all nonempty simplices of K , ordered by inclusion (which is a partial order). For example, the face-poset of the closure of {A,B,C} is the poset with the following chains:

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  9. Stanley–Reisner ring - Wikipedia

    en.wikipedia.org/wiki/Stanley–Reisner_ring

    An abstract simplicial complex Δ is called Cohen–Macaulay over k if its face ring is a Cohen–Macaulay ring. [3] In his 1974 thesis, Gerald Reisner gave a complete characterization of such complexes.