Classification of closed surfaces Surface (topology)

some examples of orientable closed surfaces (left) , surfaces boundary (right). left: orientable closed surfaces surface of sphere, surface of torus, , surface of cube. (the cube , sphere topologically equivalent each other.) right: surfaces boundary disk surface, square surface, , hemisphere surface. boundaries shown in red. 3 of these topologically equivalent each other.

the classification theorem of closed surfaces states connected closed surface homeomorphic member of 1 of these 3 families:

the surfaces in first 2 families orientable. convenient combine 2 families regarding sphere connected sum of 0 tori. number g of tori involved called genus of surface. sphere , torus have euler characteristics 2 , 0, respectively, , in general euler characteristic of connected sum of g tori 2 − 2g.

the surfaces in third family nonorientable. euler characteristic of real projective plane 1, , in general euler characteristic of connected sum of k of them 2 − k.

it follows closed surface determined, homeomorphism, 2 pieces of information: euler characteristic, , whether orientable or not. in other words, euler characteristic , orientability classify closed surfaces homeomorphism.

closed surfaces multiple connected components classified class of each of connected components, , 1 assumes surface connected.