An introduction to topology
contents (to be updated as class progresses)
- The topology of the plane (#1)
This is the first entry in an introduction to topology written for the mathclass community on LiveJournal. It is meant to be open ended: the direction it will take will depend upon the input of its readers. Readers are encouraged to actively take part in the class by asking questions and making suggestions and comments.
Very little of the material presented in this class will require any prior knowledge of topology or other areas of "advanced" mathematics. Unfamiliar notation and definitions will be avoided when it makes sense to do so, and when they are used they will be carefully explained.
The reader will not need to have any textbook. Of course, as with any subject the reader will benefit if they choose to consult a range of sources.
I will start with the same observation that L. Christine Kinsey uses to begin her text, Topology of Surfaces: topology has nothing to do with topography. So don't expect to learn too much about maps here.
So what is topology? Topology is, like geometry, a study of shape. The topologist, however, is interested in very different properties of an object's shape than the geometer. When we study geometry we are interested in distances and proportions, size and angles, lines, arcs and other curves. The topologist considers these unnecessary details, and recognizes that many fundamental similarities and differences between shapes are obscured by these measurements. For example, both a square with 400 mile long sides and a circle with a 1 cm radius are closed loops despite the fact that they have little similarity geometrically. The topologist notes that both of these shapes can be smoothly deformed into each other without tearing: we can stretch the circle outwards, pulling four points on its perimeter out to meet the corners of the square, and stretching the arcs in between them to meet the corresponding sides. Therefore, to the topologist, the two shapes are "the same". Generally, but imprecisely, two objects are topologically equivalent if there is a continuous deformation from one to the other. They may be bent, stretched, and squashed as we please.
Ok, so that's not much of a definition, but it will do for now. The best way to get a sense for what topology is is to study it, so next week we will begin by studying the topology of the plane.