Editing Surface (topology) (section)

=== Classification of closed surfaces ===
[[File:SurfacesWithAndWithoutBoundary.svg|right|thumb|200px|Some examples of orientable closed surfaces (left) and surfaces with boundary (right). Left: Some orientable closed surfaces are the surface of a sphere, the surface of a [[torus]], and the surface of a cube. (The cube and the sphere are topologically equivalent to each other.) Right: Some surfaces with boundary are the [[disk (mathematics)|disk surface]], square surface, and hemisphere surface. The boundaries are shown in red. All three of these are topologically equivalent to each other.]]

The ''classification theorem of closed surfaces'' states that any [[connected (topology)|connected]] closed surface is homeomorphic to some member of one of these three families:
# the [[sphere]],
# the [[connected sum]] of ''g'' tori for ''g'' ≥ 1,
# the [[connected sum]] of ''k'' real [[Projective plane|projective planes]] for ''k'' ≥ 1.

The surfaces in the first two families are [[orientability|orientable]]. It is convenient to combine the two families by regarding the sphere as the connected sum of 0 tori. The number ''g'' of tori involved is called the ''genus'' of the surface. The sphere and the torus have Euler characteristics 2 and 0, respectively, and in general the Euler characteristic of the connected sum of ''g'' tori is {{nowrap|2 &minus; 2''g''}}.

The surfaces in the third family are nonorientable. The Euler characteristic of the real projective plane is 1, and in general the Euler characteristic of the connected sum of ''k'' of them is {{nowrap|2 &minus; ''k''}}.

It follows that a closed surface is determined, up to homeomorphism, by two pieces of information: its Euler characteristic, and whether it is orientable or not. In other words, Euler characteristic and orientability completely classify closed surfaces up to homeomorphism.

Closed surfaces with multiple [[Connected component (topology)|connected components]] are classified by the class of each of their connected components, and thus one generally assumes that the surface is connected.