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


4.5.1 Surfaces with holes

Using this result, we can obtain the Euler characteristic of a surface with any number of holes by successively inserting the holes one at a time. For example, since a closed disc has Euler characteristic 1, it follows that a closed disc with 1 hole has Euler characteristic 0, a disc with 2 holes has Euler characteristic −1, and so on.

Problem 22

Use the above approach to determine the Euler characteristic of a torus with k holes.


Since a torus has Euler characteristic 0, it follows from Theorem 10 that a torus with 1 hole has Euler characteristic −1, a torus with 2 holes has Euler characteristic −2, and so on. In general, a torus with k holes has Euler characteristic −k.

Theorem 11

Inserting k holes into a surface reduces its Euler characteristic by k.


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