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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.

A topological surface is a Hausdorff space with a countable basis of a topology in which every point has an open neighbourhood homeomorphic to an open subset of the Euclidean plane. We use the convention that a surface has no boundary unless a surface with boundary is explicitly specified. A closed surface is compact and has no boundary. Some conventions also include connectedness in the definition; we impose it on the closed combinatorial surface below.

A closed combinatorial surface is a finite connected simplicial complex $K$ with the following properties: every simplex is a face of a simplex of dimension two, and the link of a simplex at every vertex is a cycle graph. Explicitly, for a vertex $v$ its link is the simplicial subcomplexTo say this is a cycle graph means that there are distinct vertices $w_1,\dots ,w_m$, with $m\ge 3$, and exactly the link edges $\{w_i,w_{i+1}\}$, with indices modulo $m$. In particular, every edge of $K$ belongs to exactly two triangles. The vertex-link condition also requires that the triangles around each vertex form one cyclic fan; edge incidence alone would allow singular examples.

These conditions make $|K|$ a closed surface. A point inside a triangle has a disc neighbourhood, the two triangles at an edge give a disc neighbourhood, and the cyclic fan at a vertex gives a disc neighbourhood. Finiteness gives compactness, the Hausdorff property, and a countable basis of a topology from a realization in a finite-dimensional Euclidean space. If disconnected closed combinatorial surfaces are allowed, each connected component satisfies this definition separately.