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  2. Abstract simplicial complex - Wikipedia

    en.wikipedia.org/wiki/Abstract_simplicial_complex

    Geometric realization of a 3-dimensional abstract simplicial complex. In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family.

  3. Alexander duality - Wikipedia

    en.wikipedia.org/wiki/Alexander_duality

    Let be an abstract simplicial complex on a vertex set of size . The Alexander dual X ∗ {\displaystyle X^{*}} of X {\displaystyle X} is defined as the simplicial complex on V {\displaystyle V} whose faces are complements of non-faces of X {\displaystyle X} .

  4. Triangulation (topology) - Wikipedia

    en.wikipedia.org/wiki/Triangulation_(topology)

    An abstract simplicial complex above a set is a system () of non-empty subsets such that: {} for each ;if and , then .; The elements of are called simplices, the elements of are called vertices.

  5. Vietoris–Rips complex - Wikipedia

    en.wikipedia.org/wiki/Vietoris–Rips_complex

    In topology, the Vietoris–Rips complex, also called the Vietoris complex or Rips complex, is a way of forming a topological space from distances in a set of points. It is an abstract simplicial complex that can be defined from any metric space M and distance δ by forming a simplex for every finite set of points that has diameter at most δ.

  6. Poset topology - Wikipedia

    en.wikipedia.org/wiki/Poset_topology

    The order complex associated to a poset (S, ≤) has the set S as vertices, and the finite chains of (S, ≤) as faces. The poset topology associated to a poset ( S , ≤) is then the Alexandrov topology on the order complex associated to ( S , ≤).

  7. 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.

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  9. Stanley–Reisner ring - Wikipedia

    en.wikipedia.org/wiki/Stanley–Reisner_ring

    Given an abstract simplicial complex Δ on the vertex set {x 1,...,x n} and a field k, the corresponding Stanley–Reisner ring, or face ring, denoted k[Δ], is obtained from the polynomial ring k[x 1,...,x n] by quotienting out the ideal I Δ generated by the square-free monomials corresponding to the non-faces of Δ: