# What is geometric realization?

## What is geometric realization?

Geometric realization is the operation that builds from a simplicial set X a topological space |X| obtained by interpreting each element in Xn – each abstract n-simplex in X – as one copy of the standard topological n-simplex ΔnTop and then gluing together all these along their boundaries to a big topological space.

### What is a Delta complex?

Formally, a Δ-set is a sequence of sets together with maps. with for that satisfy. whenever . This definition generalizes the notion of a simplicial complex, where the are the sets of n-simplices, and the. are the face maps.

#### What is a 2 complex?

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.

**What is simplex in math?**

In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given space. For example, a 0-simplex is a point, a 4-simplex is a 5-cell.

**Is every CW complex a simplicial complex?**

Every CW complex is homotopy equivalent to (the realization of) a simplicial complex.

## What is Delta of a set?

The Δ in set theory is the symmetric difference of two sets. In the context of elementary set theory the symbol △ usually denotes the operation of symmetric difference of two sets: if A and B are sets, A△B=(A∖B)∪(B∖A).

### What is the difference between i and J in complex numbers?

The imaginary unit or unit imaginary number (i) is a solution to the quadratic equation x2 + 1 = 0. For example, in electrical engineering and control systems engineering, the imaginary unit is normally denoted by j instead of i, because i is commonly used to denote electric current.

#### What are simplex numbers?

**What does CW complex stand for?**

closure-finite

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).

**Are manifolds CW complexes?**

Any smooth manifold admits a CW-structure. In fact it is known that any smooth manifold can be triangulated, and hence admits the structure of a simplicial complex (see example 2).