Every connected closed surface is uniquely a sphere, an orientable connected sum of tori, or a nonorientable connected sum of projective planes.Suggest a slogan
Every connected closed surface is homeomorphic to exactly one of: the sphere \(S^2\); the connected sum \(\Sigma_g = \#_{i=1}^g T^2\) of \(g\geq 1\) tori; or the connected sum \(N_k = \#_{i=1}^k \RP^2\) of \(k\geq 1\) projective planes [@Mun00]; [@Hat02]. The surfaces \(\Sigma_g\) are orientable and the surfaces \(N_k\) are not.
Under connected sum, homeomorphism classes of connected closed surfaces form a commutative monoid with identity \(S^2\), generated by \(T = T^2\) and \(P = \RP^2\) subject to the relation \(T\# P \cong P\# P\# P\); the Klein bottle is \(K\cong P\# P\) [@Mun00].
How are closed surfaces classified as a monoid on $\RP^2$, the Klein bottle and the torus?