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