Triangulation (topology)

In mathematics, topology generalizes the notion of triangulation (advanced geometry) in a natural way as follows:

A triangulation of a topological space X is a simplicial complex K, homeomorphic to X, together with a homeomorphism h:K o X.

Triangulation is useful in determining the properties of a topological space. For example, one can compute homology and Cohomology groups of a triangulated space using simplicial homology and cohomology theories instead of more complicated homology and cohomology theories.

It is known that subanalytic sets and differentiable manifolds admit a triangulation. However, some topological manifolds do not admit a triangulation (see Hauptvermutung).