Surfaces
Surfaces

This free course is available to start right now. Review the full course description and key learning outcomes and create an account and enrol if you want a free statement of participation.

Free course

Surfaces

5.2.1 Proof

We check that Tf satisfies conditions (T1)–(T3) for a topology.

Since (T1)–(T3) are satisfied, Tf is a topology on I(X).

Thus (I(X),Tf) is a topological space. We give the topology Tf a special name.

Definition

The topology Tf defined in Theorem 15 is the identification topology for X under f.

We now use F to create a topology on Y = f(X). We define a subset U of Y to be open if F−1(U) is an open subset of X.

Recall that F([x]) = f(x).

Problem 29

Show that the family TY of open sets defined in this way comprises a topology on Y.

Answer

We check (T1)–(T3).

Since (T1)–(T3) are satisfied, TY is a topology on Y.

So we have a bijection F:I(X) → Y between the topological spaces (I(X),Tf) and (Y,TY). Furthermore, from the definitions of open sets in Tf and TY, we can deduce immediately that for each U TY, F−1(U) Tf and for each V Tf, F(V) TY. Hence F is a homeomorphism between (I(X),Tf) and (Y,TY).

We have thus shown that the object created by identifying some or all of the edges of a polygon is a topological space homeomorphic to the identification space of the polygon. We now need to show that this topological space is Hausdorff and that every point has a disc-like or half-disc-like neighbourhood, to demonstrate that the space is indeed a surface.

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, find out more about the types of qualifications we offer, including our entry level Access courses and Certificates.

Not ready for University study then browse over 900 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 prospectus