CONNECTED SUM OF TWO SURFACES

The connected sum of two connected surfaces and is the surface, defined up to homeomorphism, obtained by removing an open set homeomorphic to the disk to each of the surfaces and by identifying the boundaries; notation: , or even 2S for the sum of two homeomorphic surfaces.

Examples:
- the connected sum of two spheres is a sphere: (in fact the sphere is the identity element for this operation).
- the connected sum of two real projective planes is a Klein bottle .
- the connected sum of three real projective planes (thus of a Klein bottle and a projective plane) is topologically equivalent to the sum of a torus and a real projective plane: ; we get Dyck's surface.

The classification theorem of compact surfaces without boundaries states that such a surface is either of the type (n integer), in other words, n-torus (if it has two faces), or (n positive integer), if it has only one face.
We can illustrate this theorem by saying that any compact surface without boundaries is homeomorphic to a sphere with a certain number of classic handles (tori) and Möbius handles (projective planes), with the rule that 2 Möbius handles can be changed for a classic handle, on the condition that there remains at least another Möbius handle.