## What is simplicial complexes in algebraic topology?

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.

## Is a simplicial complex a topological space?

To each simplicial complex K, one can associate a topological space called the polyhedron of K often also called the geometric realisation of K and denoted |K|. (This is essentially a special case of the geometric realisation of a simplicial sets.)

**Why are simplicial complexes important?**

Simplicial sets are very useful to algebraic topologists. Generalized triangulations are very useful to geometric topologists. Simplicial complexes are useful to combinatorialists: they are hypergraphs with a closure property.

### Is every simplicial complex a CW complex?

Notably the geometric realization of every simplicial set, hence also of every groupoid, 2-groupoid, etc., is a CW complex. Milnor has argued that the category of spaces which are homotopy equivalent to CW-complexes, also called m-cofibrant spaces, is a convenient category of spaces for algebraic topology.

### When we combine various simplices into single feature we obtain a?

When we combine various simplices into a single feature, we obtain a simplicial complex (see Figure 2 for examples).

**What is a face of a simplicial complex?**

A simplicial complex K is a collection of simplices such that (1) If K contains a simplex σ, then K also contains every face of σ. (2) If two simplices in K intersect, then their intersection is a face of each of them.

#### What is a complex topology?

A CW complex is a kind of a topological space that is particularly important in algebraic topology. The C stands for “closure-finite”, and the W for “weak” topology. A CW complex can be defined inductively. A 0-dimensional CW complex is just a set of zero or more discrete points (with the discrete topology).

#### What does CW mean in CW complex?

The C stands for “closure-finite”, and the W for “weak” topology. A CW complex can be defined inductively. A 0-dimensional CW complex is just a set of zero or more discrete points (with the discrete topology).

**Is simplex a polytope?**

A regular simplex is a simplex that is also a regular polytope. A regular k-simplex may be constructed from a regular (k − 1)-simplex by connecting a new vertex to all original vertices by the common edge length.

## Are Simplexes convex?

DEFINITION 1. The standard N-simplex is the convex hull in RN+1 of all points with all coordinates zero, except for a single coordinate with value 1. These (N + 1) points are the vertices of the simplex.

## What are the applications to cell complex?

Abstract cell complexes play an important role in image analysis and computer graphics.

**Are CW complexes Metrizable?**

It is a basic topological fact that CW-complexes aren’t typically metrizable (they must satisfy a certain local finiteness condition) and the quotient topology is to blame.