##  [Simplicial Complex](/simplicial-complex) 

  ##  [Simplicial Complex](https://natural.quantumdictionary.io/simplicial-complex-0) 

  

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**Natural & Formal Sciences Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A combinatorial structure given by a collection K of finite sets (called simplices) on a vertex set V such that every singleton vertex is in K and K is closed under taking nonempty subsets: if σ∈K and τ⊆σ then τ∈K. Simplices are organized by dimension (|σ|−1).

 

 

 

 

 





 

 



 ##  [Simplicial Complex](https://mathlogic.quantumdictionary.io/simplicial-complex-1) 

  

 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A combinatorial object consisting of a set of vertices and a collection of finite subsets (simplices) closed under taking nonempty faces: if a simplex is included then every nonempty subset of its vertices is also included.

 

 

 

 

 





 

 



 ##  [Simplicial Complex](https://puremath.quantumdictionary.io/simplicial-complex-2) 

  

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- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

An (abstract) simplicial complex K on a vertex set V is a family of finite subsets (called simplices) of V closed under taking nonempty subsets: if σ ∈ K and τ ⊆ σ then τ ∈ K. Geometric realizations glue simplices along faces to model piecewise-linear topology.