Massey Exercise 1.11.3
We can list all compact surfaces (with or without boundary) such that . Denote by the surface with discs removed. Some cells in the table have both an orientable and non-orientable possibility – non-orientable surfaces will be denoted in parenthesis. The following is a table with Euler characteristic across the top and number of boundary components down the side.
Of note is the surface , which is actually the Mobius strip. Note that none of the surfaces in this table could be homeomorphic to each other, because each differs from the rest in either Euler characteristic, number of boundary components, or both.