One Point Compactification, or “I want my locally compact space to be more compact!”
December 6, 2010
If you go to a math-savvy high school student and ask them what the complex plane looks like, they’ll probably say something like, “It’s like the xy-plane with the real values on the x-axis and the imaginary values on the y-axis.” If you go to someone studying complex analysis and ask them what the complex plane looks like, they might say something a little more surprising: “Think of it as a sphere where the top point is infinity.”
As we know, there is a big difference between a sphere and a sheet of paper — so what is going on here? The big difference is that the latter description has an actual point at infinity, and the former only has a “notion” of infinity: it keeps going, but there is no point which we label as “infinity” on our plane model. In the same respect, taking the real line, if we add a point at infinity we can think of the real line as a circle with the “top point” representing the point at infinity. Is one of these representations “better” than the other?
It depends on what you’re using it for. It’d be rather difficult to do calculus directly from a sphere (in particular, you’d have to use a bit of differential geometry) but it’s simple (or, relatively so) to do calculus on a plane. But nonetheless, there are a number of applications of “putting a point at infinity” on a space. But can we always do this?
Well. Yes. You can always stick a point on a space and make a new space from it by throwing in some open sets. But will the resulting space be useful? Not necessarily. If we want to make our resulting space compact, we need our original space to be locally compact.
There are a number of definitions of local compactness, which are all equivalent when our space is Hausdorff, but even if we don’t assume Hausdorff-ness, then we can give a sweeping definition:
Definition: A space is locally compact if every point of has an open neighborhood contained in a compact set.
Obviously, everything which is compact will be locally compact (by taking the compact set to be the whole space), so we have the following examples of locally compact spaces already: the circle, the sphere, any closed bounded set in with the standard topology, and so forth.
There are other examples of spaces which are not compact but which are locally compact. For example, the real line, the real plane, and pretty much any for finite with the standard topology (every point has an ball around it, and the closure of this ball is a compact set). We also have any discrete space is going to be locally compact (why? what are the neighborhoods in this space? what are the compact sets?), but it won’t be compact unless the entire space is finite (why?).
In fact, a lot of nice stuff is locally compact. It takes a lot to think of something which isn’t locally compact! You can prove to yourself that the lower-limit topology on isn’t locally compact (what are the neighborhoods? what are the compact subsets? can you cover with infinitely many open sets which has no finite sub-cover?) and considering the rationals as a subspace of the reals is also not locally compact.
Why did we care about this again? Oh, right! What happens when we attach a single point to our space and try to make a nice topology for this? Well, we’d like to make a topology that’s basically the same as the the one on the space, except we’d also like to include the point at infinity. Should we just link up the point at infinity up with every point? No, that would be silly. What follows is an “intuitive approach” using my intuition, so feel free to skip right to the definition if you want to develop your own intuition.
Let’s call our space and our space with the point at infinity . We can intuitively think of this question as follows: what is infinity “close to” in ? Equivalently, we can think: what is infinity “far from”, and this question turns out to be much easier to think about: if we have a compact set in our space, then this is somehow a “limited set” in the sense that we can cover it with only finitely many things. Somehow, the elements in are close together and should be far away from infinity (since, otherwise, we’d probably need more open sets to cover things!) and so we should have that infinity is not a part of this set. Thus, we can say that infinity is closer to things not in this set than things which are in this set: this is equivalent (at least, in my mind) to saying that should be an open set. Thus, by generalizing, for every compact set in our space , we should have that should be an open set in .
Even if this isn’t the intuition you use, this concept of taking compliments of compact sets turns out to be the “right” way to think about adding infinity to something. Starting with a space which is not already compact (since why would you want to compactify an already compact space?) but is locally compact, we can add a point at infinity with the open sets described above. This is called the one-point compactification of .
Definition: Given a locally compact space which is not compact, the one-point compactification of is a space where is usually called “infinity” and we endow with the open sets in the topology of as well as those sets of the form where is a compact subset of .
Below are some picture examples of what’s happening for the real line and or the complex plane.
Let’s prove the first theorem that should come to your head when you hear the phrase “this is the one-point compactification of ”, namely, let’s prove that the one-point compactification is actually compact!
Theorem: If is a locally compact (but not compact) space and is its one-point compactification, then is compact.
Proof. This isn’t too bad. Suppose we have some open cover of . Since this must cover our point at infinity, we must have that the cover includes an open set of the form for some compact set . Because our original cover covers , it also covers , and so it also covers . But because is compact, there is a finite sub-cover of our original cover. Take this finite sub-cover and that set from before, and this is a finite sub-cover of .
Note where we used locally compact here! Suppose we didn’t have a compact set around some point: we could construct a cover which had no finite subcover in ; thus, we wouldn’t have that is compact! So sad.
Some things that I like thinking about: what does the one point compactification of a finite discrete set look like? what if we try to compactify an already compact set? can we take cool looking spaces and make cool looking one-point compactifications?