Editing Surface (topology) (section)

== Closed surfaces ==
{{Redirect-distinguish|Open surface|Free surface}}

A '''closed surface''' is a surface that is [[compact space|compact]] and without [[Boundary of a manifold|boundary]]. Examples of closed surfaces include the [[sphere]], the [[torus]] and the [[Klein bottle]]. Examples of non-closed surfaces include an [[disk (mathematics)|open disk]] (which is a sphere with a [[puncturing (topology)|puncture]]), an [[open cylinder]] (which is a sphere with two punctures), and the [[Möbius strip]].

A surface embedded in [[three-dimensional space]] is closed if and only if it is the boundary of a solid. As with any [[closed manifold]], a surface embedded in Euclidean space that is closed with respect to the inherited [[Euclidean topology]] is ''not'' necessarily a closed surface; for example, a disk embedded in <math> \mathbb{R}^3 </math> that contains its boundary is a surface that is topologically closed but not a closed surface.

=== 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.

=== Monoid structure ===
Relating this classification to connected sums, the closed surfaces up to homeomorphism form a [[commutative]] [[monoid]] under the operation of connected sum, as indeed do manifolds of any fixed dimension. The identity is the sphere, while the real projective plane and the torus generate this monoid, with a single relation {{nowrap|1='''P''' # '''P''' # '''P''' = '''P''' # '''T'''}}, which may also be written  {{nowrap|1='''P''' # '''K''' = '''P''' # '''T'''}}, since {{nowrap|1='''K''' = '''P''' # '''P'''}}. This relation is sometimes known as '''{{visible anchor|Dyck's theorem}}''' after [[Walther von Dyck]], who proved it in {{Harv|Dyck|1888}}, and the triple cross surface {{nowrap|'''P''' # '''P''' # '''P'''}} is accordingly called '''{{visible anchor|Dyck's surface}}'''.<ref name="fw"/>

Geometrically, connect-sum with a torus ({{nowrap|# '''T'''}}) adds a handle with both ends attached to the same side of the surface, while connect-sum with a Klein bottle ({{nowrap|# '''K'''}}) adds a handle with the two ends attached to opposite sides of an orientable surface; in the presence of a projective plane ({{nowrap|# '''P'''}}), the surface is not orientable (there is no notion of side), so there is no difference between attaching a torus and attaching a Klein bottle, which explains the relation.

=== Proof ===
The classification of closed surfaces has been known since the 1860s,<ref name="fw">{{Harv|Francis|Weeks|1999}}</ref> and today a number of proofs exist.

Topological and combinatorial proofs in general rely on the difficult result that every compact 2-manifold is homeomorphic to a [[simplicial complex]], which is of interest in its own right. The most common proof of the classification is {{Harv|Seifert|Threlfall|1980}},<ref name="fw"/> which brings every triangulated surface to a standard form. A simplified proof, which avoids a standard form, was discovered by [[John H. Conway]] circa 1992, which he called the "Zero Irrelevancy Proof" or "ZIP proof" and is presented in {{Harv|Francis|Weeks|1999}}.

A geometric proof, which yields a stronger geometric result, is the [[uniformization theorem]]. This was originally proven only for Riemann surfaces in the 1880s and 1900s by [[Felix Klein]], [[Paul Koebe]], and [[Henri Poincaré]].