Closed combinatorial surface

A closed combinatorial surface is a finite connected simplicial complex in which every simplex is a face of a triangle and the link of a simplex at every vertex is a cycle graph. Every edge then lies in exactly two triangles, and the triangles at a vertex form a single cyclic fan. The edge-incidence condition alone is insufficient: gluing two triangulated spheres only at one vertex gives two disjoint circles in that vertex's link.

Its geometric realization is a compact surface without boundary. Triangle interiors have disc neighbourhoods, edge interiors have such neighbourhoods from the two incident triangles, and the cyclic fan gives one at each vertex. A finite realization is a Hausdorff space with a countable basis of a topology, being a finite union of geometric simplexes in Euclidean space. If connectedness is omitted by convention, require these properties separately on each connected component.