Simplicial Complex -- from Wolfram MathWorld
- ️Weisstein, Eric W.
A simplicial complex is a space with a triangulation. Formally, a simplicial complex 
in 
is a collection of simplices
in 
such that
1. Every face of a simplex of 
is in 
, and
2. The intersection of any two simplices of 
is a face of each of them
(Munkres 1993, p. 7).
Objects in the space made up of only the simplices in the triangulation of the space are called simplicial subcomplexes. When only simplicial complexes and simplicial subcomplexes are considered, defining homology is particularly easy (and, in fact, combinatorial because of its finite/counting nature). This kind of homology is called simplicial homology.
See also
Abstract Simplicial Complex, Homology, Nerve, Simplex, Simplicial Subcomplex, Simplicial Homology, Space, Triangulation
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References
Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 7, 1994.Hatcher, A. Algebraic Topology. Cambridge, England: Cambridge University Press, 2002.Munkres, J. R. "Simplicial Complexes and Simplicial Maps." §1.2 in Elements of Algebraic Topology. New York: Perseus Books Pub.,pp. 7-14, 1993.
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Cite this as:
Weisstein, Eric W. "Simplicial Complex." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SimplicialComplex.html