= Classification theorem for closed combinatorial surfaces
Every <closed combinatorial surface> has <geometric realization> <homeomorphic> to exactly one member of the following list: the <sphere> $S^2$; the <connected sum of surfaces> of $g\geq1$ <tori>; or the <connected sum of surfaces> of $k\geq1$ <real projective planes>. Every member admits a finite <triangulation>. The <orientable> family has <Euler characteristic> $2-2g$, including $g=0$ for the <sphere>; the <nonorientable surface> family has <Euler characteristic> $2-k$. Thus <orientability> and <Euler characteristic> determine the <homeomorphism> type. The first two nonorientable cases are the <real projective plane> and the <Klein bottle>. For a finite disconnected <simplicial complex> satisfying the same local conditions, the unordered list of its <connected components>' types determines its type.
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