. 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 . 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 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 , which may also be written , since . This relation is sometimes known as after Walther von Dyck, who proved it in , and the triple cross surface is accordingly called . Geometrically, connect-sum with a torus () adds a handle with both ends attached to the same side of the surface, while connect-sum with a Klein bottle () adds a handle with the two ends attached to opposite sides of an orientable surface; in the presence of a projective plane (), 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, 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 , 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 . 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. Surfaces with boundary