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# 2.2.3 Surfaces with boundary

Examples of surfaces with boundary are a cylinder and a Möbius band. Other examples are the following:

Surfaces with holes

We can obtain a surface with boundary by taking any surface without boundary and punching some holes in it by removing open discs. For example, Figure 19 shows a sphere with 3 holes. The boundary of the surface with holes consists of the boundaries of the holes.

Figure 19

Another example is a torus with holes. Figure 20 illustrates a torus with 1 hole and a torus with 3 holes. We can have a torus with k holes for any positive integer k. (Do not confuse a torus with k holes and a k-fold torus.)

Figure 20

We can also combine the notions of an n-fold torus and a torus with k holes to draw an n-fold torus with k holes, for any positive integers n and k. For example, Figure 21 depicts a 2-fold torus with 4 holes.

Figure 21

## Problem 6

Draw a 4-fold torus with 5 holes and a 5-fold torus with 4 holes. What is the boundary of each surface?

### Answer

The boundary consists of the boundaries of the 5 holes.

The boundary consists of the boundaries of the 4 holes.

Paper surfaces

Other surfaces can be drawn as if they were made from thin strips of material, such as paper. The surface consists of one side of the paper only. Two such surfaces are shown in Figure 22. Notice how some parts of a surface can cross over, or under, other parts.

Figure 22 Two paper surfaces
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