Following infinite sets, the next topic covered in the textbook is topology. The course notes skip straight over topology and move on to sequences. Comparing the two, it looks like the course notes limit discussion of sequences to Euclidean spaces, while the textbook discusses sequences in metric spaces in general. Since the coverage of the topic is more general, the book needs to develop more of a theoretical framework.
My attitude is that as long as it's in the book, I might as well take a look. There are some caveats for my posts here. I'm less familiar with topology than I am with previous topics, so I'm more dependent on the book for making sense of the material. Also, on previous topics, I was able to compare the notes and the book, so I got two shots at understanding, and now I only have one.
The book's presentation of this material is very dense. It could be that the book expects this is review and is covering it quickly. Or it may consider this background to the later material, so while it may be new to the reader, the book doesn't want to spend more time on it than necessary. Or the book could always be this dense, and I just haven't noticed until now.
In any event, the presentation of the material in the book isn't very friendly to comprehension. It starts with a full page of definitions of terms, and then starts cranking through theorems expecting an understanding of those terms. My goal is to restructure the material, giving more of a chance to work with each concept before moving on to the next. This may make things more comprehensible. Or it may make things a giant mess.
The first topic is actually a bit of review. I've already mentioned metric spaces, but now I'm going to take another look. A metric space is a set of points with a distance function defined between pairs of points. Given any two points in the space, the distance function gives us a real number which specifies the distance between them. In order to qualify as a metric space, the distance function must have three properties.
- For any two points p and q, d(p,q)≥0. Furthermore, d(p,q)=0 if and only if p=q. The distance from any point to itself is always 0. The distance from any point to any other point is always a real number greater than 0.
- For any two points p and q, d(p,q)=d(q,p). Living outside of Boston, I'm well aware that driving directions from one place to another can be radically different depending which way you're going. One way streets make a mess of trip planning, and sometimes the route to a destination is significantly shorter than the route back. In a metric space, the distance is always the same, regardless of which direction you are traveling in.
- For any three points p, q, and r, d(p,q)≤d(p,r)+d(q,r). (The triangle inequality) I was about to say that the distance function doesn't care which path you take, as long as the distance is the same, but that's not completely accurate. d(p,q) is the direct distance between p and q. If you start from p, go to some other point r, and then proceed to q, the combined distance of that trip is always equal to or greater than the direct distance. You can't take a shortcut by deliberately going through another point. (Boston geography is looking less and less like a metric space all the time.)
This definition means that sets with the distance functions we expect to work are metric spaces. In particular, for the set of real numbers, the distance function d(p,q)=|p−q| defines a metric space. For ordered pairs of real numbers, or higher dimensional Euclidean spaces, the distance as measured as the length of a vector between the two points defines a valid metric space.
However, different distance functions for the same set of points can also define a valid metric space. For example, one metric space with more relevance to New York than to Boston uses taxicab geometry to find distances on the 2-d plane. First you have to travel right or left, then you can travel up or down. In formulas, if p=(p1,p2) and q=(q1,q2), the taxicab distance is d(p,q)=|p1−q1| + |p2−q2|.
Also, different sets of points with the same distance function are distinct metric spaces. I expect that comparing the space of rational numbers to the space of real numbers, both with d(p,q)=|p−q|, will help clarify some of the properties of metric spaces.