Editing Torus (section)

== Genus ''g'' surface ==
{{Main|Genus g surface}}
In the theory of [[surface (topology)|surface]]s there is a more general family of objects, the "[[genus (mathematics)|genus]]" {{math|''g''}} surfaces. A genus {{math|''g''}} surface is the [[connected sum]] of {{math|''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 {{math|''g''}} surface resembles the surface of {{math|''g''}} doughnuts stuck together side by side, or a [[sphere|2-sphere]] with {{math|''g''}} handles attached.

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

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

{| class=wikitable
|- align=center
|[[File:Double torus illustration.png|160px]]{{br}}[[double torus|genus two]]
|[[File:Triple torus illustration.png|240px]]{{br}}[[triple torus|genus three]]
|}