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# 2.2.1 Surfaces without boundary

Examples of surfaces without boundary are a sphere and a torus. Other examples are the following:

n-fold toruses

Figure 13 depicts a 2-fold torus and a 3-fold torus, with two and three rings respectively. An n-fold torus, for any positive integer n has n rings. (A 1-fold torus is simply a torus, as in Figure 3.)

Figure 13

Toruses do not have to be considered as ‘n-ringed doughnuts’ as in Figure 13; they can also usefully be described as ‘spheres with n handles’, as Figure 14 illustrates.

Figure 14 Two different representations of a 2-fold torus

In fact, any homeomorphic representation will do, as we shall see in Section 2.4.

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