Surfaces:

The sphere $S^2$, torus, Klein's bottle and projective plane are the four basic examples of a class of spaces called surfaces. We shall not formally define a surface but provide one more example namely, the double torus. Roughly the double torus is obtained by taking two copies of the torus and cutting out a little disc form each of them so as to obtain a pair of tori each with a boundary. One then glues these boundaries together to obtain a double torus.

Figure: Double torus as a connected sum

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Analytically the double torus is the identification space obtained by identifying pairs of opposite sides of an octagon according to the following scheme.

Figure: Double torus

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Obviously the process can be generalized and one can obtain for instance a triple torus by identifying pairs of opposite sides of a twelve sided polygon. The classification of surfaces forms an important chapter in topology and we refer to the book of [11].