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 Edge identifications

5.1 Identifying edges of a polygon

In this section, we revisit the construction of surfaces by identifying edges of polygons, as described in Section 2. Recall that, if we take any polygon in the plane and identify some of its edges in pairs, then we obtain a surface. When specifying how a given pair of edges is to be identified, we choose one of the two possible relative orientations of these edges.

In diagrams we represent such a rule for identifying edges by labelling the edges with letters and arrowheads, as in Figure 101: a pair of edges is identified if each carries the same letter, and the direction of identification is indicated by the arrowheads. We refer to an edge labelled with an arrowhead as a directed edge. Any assignment of letters and arrowheads to edges in which a particular letter appears on exactly two directed edges defines a rule for identifying edges of a polygon.

Figure 101

To prove that we do obtain a surface when we identify edges of a polygon in this way, we need to show how to make the resulting set into a topological space. We use a topology called the identification topology. We explore this topology and its consequences in this section.

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