Last week we introduced the idea of compact sets, and we started to explore the properties of compact sets. There's more to say about compact sets, and I expect to spend most of this week discussing them. However, before I do that, I want to explore some ideas I've been kicking around comparing metric spaces of real numbers and metric spaces of rational numbers.
Most of my examples when discussing metric spaces for the past couple of weeks have been based on spaces of real numbers, although that has been implied in some cases. My examples involving a piece of paper and sets marked out as squares or circles on the paper are effectively a visualization of a 2 dimensional Euclidean space. Every point on the page corresponds to an ordered pair of real numbers, and the distance function for the space is the straight line distance between the two points.
(An aside which I'm not going to develop right now, but maybe I should someday: Roll the piece of paper into a cylinder, and measure the distance as the straight line distance through the cylinder, rather than along the surface of the cylinder. Is this a valid metric space? I assume it is, but I'd have to actually work through the math to make sure. Anyway, back to the topic at hand.)
The other space I've used in various examples is the real number line, which you could also call the 1 dimensional Euclidean space. I think I've also referred a couple of times to the set of rational numbers, with distance measured as the absolute value of the difference. This space is of course a subspace of the real number line, but because it has holes, it has some different properties.
I want to compare some sets on the real number line with sets on the rational number line, so we can see how they behave differently. Let's start on the real line with the set of real numbers greater than −2 and less than 2. This is an open set. If it included the endpoints, i.e., the set of real numbers greater than or equal to −2 and less than or equal to 2, it would be a closed set. Neither set is a valid set in the space of rational numbers, since both sets include irrational numbers.
On the other hand, we can look, on the rational number line, at the set of rational numbers greater than −2 and less than 2. On the rational number line, this is an open set. However, on the real number line, this set is not open. The holes in the set guarantee that in the real number space, every neighborhood of every point in a set of rational numbers is missing some points, so no set of rational numbers can ever be an open set in the real space. (Note that the set is also not closed in the real space. There is one sense in which open and closed are opposite concepts, and another in which they are not.) This gets back to the previous discussion about how sets are only open or closed relative to the space in which they are defined.
Likewise, the set of rational numbers greater than or equal to −2 and less than or equal to 2 is a closed set in the rational number line space, but is not closed in the real number line space. Closed sets are slightly different than open sets, however. It is possible for a set to be closed on both the rational number line and on the real number line. For example, a set of isolated points can have no limit points, and will therefore be closed in either space.
Okay, now let's look on the rational number line at the set of numbers which is greater than −√2 and less than √2. The critical thing is that since √2 is not a rational number, this set is both open and closed on the rational line. It should be obvious that it is open. It is closed because every rational number which is a limit point of the set is a member of the set. Since √2 is not a rational number, it does not exist in the rational number space, and therefore cannot be a limit point of the set. Once again, of course, this set is neither closed nor open on the real line. Sets of real numbers between −√2 and √2 can be either open or closed, depending on whether the set includes the end points. If the set includes one endpoint but not the other, it's neither open nor closed, but it can't be both open and closed.
Bringing this back to more recent topics, let's look at whether the set of rational numbers greater or equal to −2 and less than or equal to 2 is compact. We know that if it is compact in one space, it is compact in every space. We also know that if it is compact, it is closed. Since it is not closed on the real number line, it can't be compact, but I want to take a look at the reason why.
Let's look at the set on the real number line. Once again, we're going to make use of our favorite irrational number, √2. We are going to build an open cover of the set which does not cover √2, and we are going to build it specifically so that no finite subcollection of the open cover is also an open cover.
We are going to define the open cover inductively. We are going to start by creating a neighborhood of the number 2. For the radius of the neighborhood, we will pick some rational number less than half the distance from 2 to √2. We can try to be greedy and pick a number which is close to half the distance, but we can pick a smaller number too. It will work out the same. Let's pick 1/4. Now, the point 2 − 1/4 is a rational number, and is not included in the neighborhood of 2. (Neighborhoods are open sets, and don't include points on the edge of the set.) However, the neighborhood includes all the points greater than 1 3/4. So let's do it again. Set 1 3/4 as the center of a neighborhood and set the radius to a rational number less than half the distance to √2.
It should be clear that after any finite number of neighborhoods is created, there will be some maximum rational number greater than √2 which is not in any of the neighborhoods, but for which all rational numbers greater than that number are in one of the created neighborhoods. (If the words don't make sense, draw a picture and it should be clear.) However, for any rational number greater than √2, there is also a finite number of neighborhoods after which the number will be included in one of the sets.
We could build a similar collection of sets coming from the other direction, but we don't even have to. We can just take the neighborhood of −2, with radius 2+√2, and we'll pick up all of the rational numbers greater or equal to −2 and less than √2. So now we have an open cover of the set of rational numbers greater than or equal to −2 and less than or equal to 2. This collection of sets was constructed to exclude √2, which is not a problem because √2 is not rational. But the collection was also constructed to be infinite, and so that no finite subcollection would be an open cover of the set of rational numbers greater than or equal to −2 and less than or equal to 2. Therefore, the set of rational numbers greater than or equal to −2 and less than or equal to 2 is not a compact set.
Note that the same definitions of neighborhoods, each with the same center and radius, also builds an open cover of the set on the rational line. There's a slight conceptual difference, because the neighborhoods on the real line are of all real numbers, but the neighborhoods on the rational line are just of rational numbers. It also doesn't quite make sense to say that the open cover on the rational line excludes √2, since √2 is not a point on the rational line, but it is true that all of the neighborhoods with centers greater than √2 overlap, and all of the neighborhoods (if we constructed more than one) with centers less than √2 overlap, but no neighborhood with a center greater than √2 overlaps a neighborhood with a center less than √2. And this is a topic which leads us to tomorrow's digression.