Surfaces

Start this free course now. Just create an account and sign in. Enrol and complete the course for a free statement of participation or digital badge if available.

Free course

# 2.2.4 Boundary numbers

In a surface with boundary, the boundary may consist of more than one piece. For example, the boundary of a cylinder, or of a torus with two holes, consists of two pieces, as shown in Figure 23.

Figure 23

The separate pieces forming the boundary of a surface S are called the boundary components of S. The number of boundary components is the boundary number of S, denoted by β. Thus β = 2 for each of the surfaces in Figure 23.

For a surface without boundary, there are no boundary components, so β = 0.

(β is the Greek letter beta.)

## Problem 7

For each of the surfaces in Figure 24, write down the boundary number β.

(a) β = 1; (b) β = 4; (c) β = 1

To find the boundary numbers of paper surfaces, it is convenient to use coloured pens to trace around the boundary. Doing this, we find that the two boundary numbers of the surfaces in Figure 22 are β = 4 and β = 3, as shown in Figure 25.

## Problem 8

Find the boundary number β of each of the paper surfaces in Figure 26.

Figure 26

The boundary number β is one of the three characteristic numbers that we use to classify compact surfaces.

M338_1

### Take your learning further

Making the decision to study can be a big step, which is why you'll want a trusted University. The Open University has 50 years’ experience delivering flexible learning and 170,000 students are studying with us right now. Take a look at all Open University courses.

If you are new to University-level study, we offer two introductory routes to our qualifications. You could either choose to start with an Access module, or a module which allows you to count your previous learning towards an Open University qualification. Read our guide on Where to take your learning next for more information.

Not ready for formal University study? Then browse over 1000 free courses on OpenLearn and sign up to our newsletter to hear about new free courses as they are released.

Every year, thousands of students decide to study with The Open University. With over 120 qualifications, we’ve got the right course for you.

Request an Open University prospectus371