A **simplicial complex** is a set of simplices that satisfies the following conditions:

- 1. Any face of a simplex from is also in .
- 2. The intersection of any two simplices is a face of both and .

Note that the empty set is a face of every simplex. See also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry.

A **simplicial k-complex** is a simplicial complex where the largest dimension of any simplex in equals

*k*. For instance, a simplicial 2-complex must contain at least one triangle, and must not contain any tetrahedra or higher-dimension simplices.

A **pure** or **homogeneous** simplicial *k*-complex is a simplicial complex where every simplex of dimension less than *k* is a face of some simplex of dimension exactly *k*. Informally, a pure 1-complex "looks" like it's made of a bunch of lines, a 2-complex "looks" like it's made of a bunch of triangles, etc. An example of a *non*-homogeneous complex is a triangle with a line segment attached to one of its vertices.

A **facet** is any simplex in a complex that is *not* a face of any larger simplex. (Note the difference from a "face" of a simplex). A pure simplicial complex can be thought of as a complex where all facets have the same dimension.

Sometimes the term *face* is used to refer to a simplex of a complex, not to be confused with a face of a simplex.

For a simplicial complex embedded in a *k*-dimensional space, the *k*-faces are sometimes referred to as its **cells**. The term *cell* is sometimes used in a broader sense to denote a set homeomorphic to a simplex, leading to the definition of cell complex.

The **underlying space**, sometimes called the **carrier** of a simplicial complex is the union of its simplices.

Read more about Simplicial Complex: Closure, Star, and Link, Algebraic Topology, Combinatorics

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### Famous quotes containing the word complex:

“All of life and human relations have become so incomprehensibly *complex* that, when you think about it, it becomes terrifying and your heart stands still.”

—Anton Pavlovich Chekhov (1860–1904)