compactification............
part 11 of topology (toc)
So in my excitement to get to algebraic topology (because come on it's cool), I totally forgot to finish off the non-algebraic part of the class. I mean, granted, this stuff isn't so bad either, but you just can't compare it to homotopies, man.
Anyways, let me present you guys with a fun math ad-lib!
A space, X, is locally asdf if for all points x ∈ X, and all neighbourhoods U of x, there is a asdf neighbourhood V of x with x ∈ V ⊂ U. Let's do an example, since (as much as I wish it were) "asdf" is not a topological property.
A space, X, is locally connected if for all points x ∈ X, and all neighbourhoods U of x, there is a connected neighbourhood V of x with x ∈ V ⊂ U. This definition works for locally path-connected if you insert path- in front of every instance of connected.
Basically what this definition is saying is that for every neighbourhood of a point, it contains a smaller neighbourhood that has the property you are interested in. We did not use locally connected at any point in the class, but we did use locally path-connected. Remember when I was rambling about Counterexamples in Topology and the topologist's sine curve as an example of a space whose connected and path-connected components differed? It turns out that if a space is locally path-connected, then its connected components and path-connected components will coincide. In particular, a space that is connected and locally path-connected is just plain old path-connected.
Anyways, immediately after introducing that math ad-lib, I'm going to shoot it into the ground with the definition of local compactness. Local compactness does not follow this definition recipe, so here is the actual definition: a space, X, is locally compact if for every x ∈ X, there exists a compact subspace, C, containing a neighbourhood of x. Notice how this definition is not a "for all" construction, but a "there exists" construction. Basically, it is asking for a "fat" compact area around x; "fat" in the sense that it needs to contain a neighbourhood of x, so it cannot just be the point {x} itself.
I bring up locally compact only because we used it to talk about one-point compactifications; specifically the Alexandroff extension. Let me motivate it by returning to the simple example of [0, 1) in the real euclidean line. This subspace is not compact, because it does not contain the endpoint 1. If we just add in this extra point, we have made it compact, or compactified it in a way. Thinking about compactifications is kind of nice because, as I said in my real analysis definition dump what seems like fifty years ago, compact spaces are a "small" and manageable kind of infinite space.
Generals first, then specifics. An embedding of a space X, into a space Y, so that X is dense in Y and Y is compact is a compactification of X. Another way of saying X is dense in Y is to say that the closure of X is exactly Y. Unfortunately, this general definition doesn't really give us any clue on how to go about finding such an embedding. As I understand, this problem is largely a pretty hard one. But there are some convenient constructions have smart people have figured out for us.
Which brings me back to the one-point compactification. To qualify for the one-point compactification, X needs to be locally compact and Hausdorff (but obviously not compact, because then the compactification discussion would be a moot point). The one-point compactification of such a space is usually called X*, and set-wise is defined as X ∪ {∞} (you could really use any symbol for the extra point, but I will explain why we choose ∞ in a moment). The topology on X* consists of all sets U which were originally open in X, as well as the addition of all subsets V containing {∞} such that X* - V is closed and compact in X.
What this compactification is doing is essentially leaving X the way it is, but throwing in one more point and adding some open neighbourhoods around that point. Returning back to my example, [0, 1] is the one-point compactification of [0, 1). I say "the" one-point compactification because it actually turns out that if there are two such compactifications, they are homeomorphic (where the homeomorphism restricted to X is just the identity). Basically, we rename the extra point in the two compactifications so they are the same.
Another example of the one-point compactification that is more visual in nature is that of the real line. The real line is not compact, since it is not bounded, but it is locally compact, since you can always pick a closed interval around any point. As such, we can construct the one-point compactification by adding in ∞. The one point compactification of the real line is homeomorphic to a circle: basically, you take the line and bend it into a circle, and add one point that joins both ends. Now, it makes sense to call the extra point ∞, and that is a convention that we end up saving for all other one-point compactifications too. If it's a little trippy for you to think about the infinite real line, then just think about (0, 1) instead. It is a line segment that is open on both ends; imagine bending it around so the open ends are next to each other, and then just inserting one point to fill in the empty spot. Ta-da! Instacompactification, ready for your enjoyment.
We briefly dabbled in the Stone-Cech compactification too, but my postdoc kind of sheepishly introduced it mostly for future reference purposes. I guess I'll just leave the name here, but I am definitely not qualified or knowledgeable enough to really talk about it too much. (I do not talk the talk when I cannot walk the walk, yo.)
Sorry for the brief detour from algebraic topology... but guys, seriously, I really hate commutative diagrams.
