Math in Plain English: Topology I – Metric Spaces I

The series of posts that I am starting with this is an attempt to explain so-called higher-level mathematics to people who don’t have strong mathematical backgrounds. The ultimate goal will be to explain my area of study, operator algebras, which seems like an extraordinary task since the subject requires so much just to get started with what they are. In addition to the technical challenges, the goal here isn’t to build up superficial knowledge, like what the definition of a C^{\ast}-algebra is without knowing the context for why it is what it is. When trying to motivate mathematics, it seems as though the best option is to look at why they were first studied. So I will begin not quite at the very beginning, but an important historical result and for that I need to first talk about topology.

Topology, in oversimplifying terms, is the subject of qualitative features of geometric objects. Here there are lots of problems already. What do I mean by qualitative? What do I mean by geometric object? But what I mean is a bit too abstract in the fullest generality that topology often begins. So instead I will start, like most analysts, with a more quantitative and concrete notion of metric space.

A metric space, in short, is a list of the possible places (called points) to be and the distances between each place. To put this in a bit more formal language.

Definition. A metric space is a collection of points along with a notion of distance from one point to the other satisfying the following conditions:

  1. The distance between any two points always remains the same. For example, the Earth and Mars have different distances from each other depending on the time, so we’re ruling out this sort of behavior. Our points are in that sense static, they don’t move around.
  2. The distance from one point to another is always a positive number or zero.
  3. The distance from a point to itself is zero and if the distance from one point to another is zero, then the two points are the same.
  4. There are no one-way streets, where it might be easier to go one direction than it is in the other. The distance from a starting point to an ending point is the same as the distance from end to start. So we can drop the “from” and “to” language and talk about the distance between two points.
  5. Pit stops won’t make the distance any shorter. If I go by someplace else first, the distance will be at least as long as compared to going straight to my destination.

So this definition might feel cumbersome, but it has an unambiguity that’s quite nice. For now, our class of geometric objects are (metric) spaces, and we can tell when we have one such space when all these conditions hold. So let’s take some examples:

  1. Take a globe. Consider the points to be all the major cities that are labeled and the distance to be their distance as we measure it: the length of the shortest line we draw between the cities.
  2. Take the globe again. This time take every possible position on the globe with the same distance as before. So we see that we can get different spaces by considering more or fewer points.
  3. Take the globe once again. Consider the points to be all the major cities. The distance this time is the shortest time it would take to travel between the two cities. Here we might run into the problem that going from one city to another might not take the same time as the other way around. So here our points did not change, but our sense of distance changed. So none of these three spaces are considered the same, even though the points in this example are the same as in the first example.
  4. Consider a country with a major railway. The points this time are the stops and the distance is the shortest distance you have to travel to get from one stop to the other. An example of this inspires a mathematical abstraction known as the “British railway” or “French railway” metric.
  5. So despite usually associating these with places and distances, one of things about mathematics is that these notions can generalize to ideas that have nothing to do with physical distance. So consider points to be English words and the distance is 1 divide by the first place where the two words differ and the distance is 0 if they are the same word. So “about” and “abacus” have a distance of 1/3 since “ab” agree but they disagree on the third letter. For another example, “demon” and “demonstration” have a distance of 1/6, since the word “demon” ends before “demonstration” does.

So the notion of metric space gives us a starting point about talking about this qualitiative study of geometry, which I will go into more detail later.


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