Editing Triangulation (topology) (section)

== Definition ==
A triangulation of a topological space <math>X</math> is a [[homeomorphism]] <math>t: |\mathcal{T}|\rightarrow X</math> where <math>\mathcal{T}</math> is a simplicial complex. Topological spaces do not necessarily admit a triangulation and if they do, it is not necessarily unique.

=== Examples ===

* Simplicial complexes can be triangulated by identity.
* Let <math>\mathcal{S}, \mathcal{S'}</math> be as in the examples seen above. The closed unit ball <math>\mathbb{D}^3</math> is homeomorphic to a tetrahedron so it admits a triangulation, namely the homeomorphism <math>t:|\mathcal{S}| \rightarrow \mathbb{D}^3</math>. Restricting <math>t</math> to <math> |\mathcal{S}'|</math> yields a homeomorphism <math> t':|\mathcal{S}'| \rightarrow \mathbb{S}^2</math>. [[File:Sphere triangulated2.png|thumb|none|200px|The 2-dimensional sphere and a triangulation]]
* The [[torus]] <math>\mathbb{T}^2 = \mathbb{S}^1 \times \mathbb{S}^1</math> admits a triangulation. To see this, consider the torus as a square where the parallel faces are glued together. A triangulation of the square that respects the gluings, like that shown below, also defines a triangulation of the torus. [[File:Torus paths2.png|thumb|none|329px|A two dimensional torus, represented as the gluing of a square via the map g, identifying its opposite sites]]
* The [[projective plane]] <math>\mathbb{P}^2</math> admits a triangulation (see CW-complexes)
* One can show that [[differentiable manifold]]s admit triangulations.<ref name="On C1-Complexes">{{citation|surname1=J. H. C. Whitehead|periodical=Annals of Mathematics|title=On C1-Complexes|volume=41|issue=4|at=pp.&nbsp;809–824|issn=0003-486X|date=1940|doi=10.2307/1968861
|jstor=1968861 }}</ref>