stylistic:tone, structural:expansion, linguistic:paraphrasing, stylistic:style_transfer, linguistic:translation
Genus g surface

In the theory of surfaces there is a more general family of objects, the " genus " g surfaces. A genus g surface is the connected sum of g two-tori. (And so the torus itself is the surface of genus 1.) To form a connected sum of two surfaces, remove from each the interior of a disk and "glue" the surfaces together along the boundary circles. (That is, merge the two boundary circles so they become just one circle.) To form the connected sum of more than two surfaces, successively take the connected sum of two of them at a time until they are all connected. In this sense, a genus g surface resembles the surface of g doughnuts stuck together side by side, or a 2-sphere with g handles attached.

As examples, a genus zero surface (without boundary) is the two-sphere while a genus one surface (without boundary) is the ordinary torus. The surfaces of higher genus are sometimes called n -holed tori (or, rarely, n -fold tori). The terms double torus and triple torus are also occasionally used.

The classification theorem for surfaces states that every compact connected surface is topologically equivalent to either the sphere or the connect sum of some number of tori, disks, and real projective planes.