summands, minus two: The sphere S is an identity element for the connected sum, meaning that . This is because deleting a disk from the sphere leaves a disk, which simply replaces the disk deleted from M upon gluing. Connected summation with the torus T is also described as attaching a "handle" to the other summand M. If M is orientable, then so is . The connected sum is associative, so the connected sum of a finite collection of surfaces is well-defined. The connected sum of two real projective planes, , is the Klein bottle K. The connected sum of the real projective plane and the Klein bottle is homeomorphic to the connected sum of the real projective plane with the torus; in a formula, . Thus, the connected sum of three real projective planes is homeomorphic to the connected sum of the real projective plane with the torus. Any connected sum involving a real projective plane is nonorientable. Closed surfaces A closed surface is a surface that is compact and without boundary. Examples of closed surfaces include the sphere, the torus and the Klein bottle. Examples of non-closed surfaces include an open disk (which is a sphere with a puncture), a 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 that contains its boundary is a surface that is topologically closed but not a closed surface. Classification of closed surfaces The classification theorem of closed surfaces states that any 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 planes for k ≥ 1. The surfaces in the first two families are 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