= 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>.
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