Surfaces
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

Surfaces

5.3 Neighbourhoods

We know that a polygon X is a surface and so each point x in X has a disc-like or half-disc-like neighbourhood. We shall show that a map f that identifies edges of a polygon to create an object Y automatically creates corresponding disc-like or half-disc-like neighbourhoods of each point y = f(x) of Y.

If x is in the interior of X, there is no difficulty: the point x has a disc-like neighbourhood U which is mapped by f to a disc-like neighbourhood f(U) of y = f(x) in Y.

If x lies on the boundary of X, there are three cases to consider:

  • x lies on an edge that is identified with another edge, but not at a vertex of that edge;

  • x lies at a vertex of an edge that is identified with another edge;

  • x lies on a part of the boundary that is not identified with another part.

In each case, x has a half-disc-like neighbourhood in X. Some examples will help us determine which happens to these half-disc-like neighbourhoods after identification takes place.

M338_1

Take your learning further371

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 courses372.

If you are new to university level study, we offer two introductory routes to our qualifications. Find out Where to take your learning next?373 You could either choose to start with an Access courses374or an open box module, which allows you to count your previous learning towards an Open University qualification.

Not ready for University study then browse over 1000 free courses on OpenLearn375 and sign up to our newsletter376 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