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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.
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.
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.
Formally, a Δ-set is a sequence of sets {} = together with maps : + for each and =,, …, +, that satisfy + = + whenever <.Often, the superscript of is omitted for brevity.. This definition generalizes the notion of a simplicial complex, where the are the sets of n-simplices, and the are the associated face maps, each mapping the -th face of a simplex in + to a simplex in .
In topology and combinatorics, it is common to "glue together" simplices to form a simplicial complex. The geometric simplex and simplicial complex should not be confused with the abstract simplicial complex, in which a simplex is simply a finite set and the complex is a family of such sets that is closed under taking subsets.
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Using simplicial homology example as a model, one can define a singular homology for any topological space X. A chain complex for X is defined by taking C n to be the free abelian group (or free module) whose generators are all continuous maps from n-dimensional simplices into X. The homomorphisms ∂ n arise from the boundary maps of simplices.
Download as PDF; Printable version; In other projects ... Hyperboloid of one sheet; Hyperboloid of two sheets; ... Abstract Polytopes 1: 1 line segment: 0: 1: 0: 0: 0 ...
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1747 Olentangy River Rd, Columbus, OH · Directions · (614) 299-9425