Many people first learn about topology as a generalization of a metric space. We can observe that the open sets of a metric space obey three rules:
- The entire space is open, as well as the empty set
- The intersection of finitely many open sets is open
- The arbitrary union of open sets is open
These are the axioms of a topology. But what if we could motivate these without reference to the behavior of a metric space? This is my attempt to do so by making an argument about what a “neighborhood” should be, and trying to argue that the definitions of a topology capture the behavior we want.
Definitions are meant to emulate observed or desired behavior through mathematical formalism. So the first question to ask would be, “what behavior are we trying to capture?” I think a reasonable (or at least sufficient for now) answer would be homeomorphism. We want a way to say that two things have “the same shape,” which is strict enough to capture properties we want, such as intuitive (non-formal for now) notions of “connectedness,” or “compactness,” but loose enough that geometric information such as size, length, curvature, etc. don’t come into play.
The basic unit of information we will be working with should be able to capture the “topological” information around each point. One way is to talk about “neighborhoods” around each point. Informally, if you google “neighborhood definition,” one result will say “the area surrounding a particular place, person, or object.” So we want to think about an object by using information about what it means for two parts of it to be “in a neighborhood” of each other. By talking about these neighborhoods, perhaps we will be able to capture the information we want by specifying which points are “close to” each other in this sense.
So, with this vague starting point, we next ask ourselves: “without knowing anything else about neighborhoods, what rules should they follow?”
First, we might make the very obvious (but very necessary) observation that a set which can be called a neighborhood of a point must certainly contain that point. That’s a starting point. Closely following this would be the assertion that every point ought to have a neighborhood, even if it’s all alone.
A starting point for a definition would be a way to specify neighborhoods. Our rough draft definition will be:
A topology is a set , equipped with a function , which picks out for each point a collection of subsets of , which are to be designated that points “neighborhoods.” What rules should this function follow?
Next, we want to think about which sets containing a point should or shouldn’t be a neighborhood of a point. Without reference to any other behavior, one immediate property of sets we can establish is whether or not that set contains a neighborhood of that point. Heuristically, neighborhoods should be somewhat local, so we don’t care how a neighborhood of point treats point . Formally, we can say that being a neighborhood and containing a neighborhood are the same.
In other words, for any point , if , and , then .
As a sidenote, I think that at this point we have the power to tell if a point is “isolated” from every other point. Specifically, a point will be called an isolated point if . The local behavior around that point is the same as the behavior at only that point: nothing exists immediately around it.
Next, we ask ourselves “how should two neighborhoods of a point interact?” Well, if two sets each contain “an area around a point,” then their intersection should, too. Morally, the property of being “around a point” means that if the intersection of two sets does not contain “an area around the point”, that should mean at least one of those two sets was not a neighborhood. This will be our third axiom:
If , then .
This is a nice list of axioms so far. Is there anything else we want?
Well, we wanted to consider “closeness.” So, we should figure out what it should mean for two points to be “in a neighborhood of each other.” Earlier, I said that a point is isolated if it has a neighborhood consisting only of itself. So, if a point is not isolated, it should at least matter a little bit how the points around it behave. After all, if a set is a neighborhood of , shouldn’t have some neighbors?
Let’s say that are “neighbors in ” if . Clearly this is an equivalence relation. Consider some neighborhood of , and let denote the equivalence class of elements which are “neighbors in ” to . Everyone here are neighbors, so shouldn’t this set be a neighborhood? Formally, we say:
If , then there exists such that for all , we have .
To conclude this section, we state all the axioms we came up with again:
A on a set , is a function which satisfies the following axioms:
- If , then
- If , and , then .
- If , then .
- If , then there exists an such that for any , .
According to wikipedia, these axioms are due to Felix Hausdorff, whose eponymous spaces are very good indeed. We then come to the following important definition:
Given the above information, a subset is called if it is a neighborhood of every point it contains. That is, if for all , we have .
Thanks to my friend @kimmellionaire on Twitter, who pointed out that the above is not quite sufficient! We also need the following axiom:
- Every point has at least one neighborhood. That is, for every ,
Without this axiom, we wouldn’t necessarily have that the entire space is open. Can you figure out why?
Here is an example of how this captures behavior we want. We can tell if two objects are completely distinct – if we have two open sets which are completely disjoint, then anything which lies in solely in one is surely distinct from anything which lies solely in the other. After all, you can go anywhere in the first without being in a neighborhood of the second, and vice versa. This suggests the following definition:
A topological space is called if there exist two disjoint nonempty open subsets whose union is . Conversely, it should be called if it is not disconnected.
We can also formalize the notion of “compactness” in a useful way. Informally, compactness should mean that it is in some sense “small,” but while containing a lot. It should be some more general version of being finite, as that is a notion of “smallness” which is easily distinguishable from things which are not small.
We capture this information as follows. Suppose that we have a topological space , and an open cover . That is, a such that every set in is open, and the union of all sets in is the entire space . If something were to be “compact,” then surely we would not require many sets to cover it. The notion of smallness we have is finiteness, so maybe the thing that should be small is any cover. Or at least, any cover should really just be a small cover with some added bonus sets. We arrive at the following definition:
A topological space is called if for any open cover , there exists a finite such that is also an open cover.
In the next post, I will try to justify why the definition of continuty we have is correct. Until then, your exercise is to show that this definition of a topology via neighborhoods is equivalent to the definition via open sets. That is, given a neighborhood function , we can recover a unique collection of open sets , and given that same collection , we can recover a unique neighborhood function , which will be exactly the one we started with.