Editing Surface (topology) (section)

=== 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.