Triangulation

Remarks

Let $K$ be the simplicial complex of all proper faces of a $3$-simplex in $\R^3$, with $4$ vertices, $6$ edges and $4$ triangles. Radial projection from the barycentre onto the sphere centred there is a homeomorphism $\abs{K} \to S^2$, so $K$ with this map is a triangulation of $S^2$.

The CW structure of $S^2$ with one $0$-cell and one $2$-cell is not a triangulation. Its $2$-cell is glued to the point along its whole boundary circle, while in a simplicial complex each triangle has three distinct vertices and three edges.

Some authors allow only finite or locally finite complexes $K$, and some require $h$ to be piecewise linear in a given structure. Every smooth manifold is triangulable, while some topological $4$-manifolds, such as the $E_8$ manifold, are not.