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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.
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 , ≤).
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} .
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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.
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.
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.
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 theory.