As we go on, we’re going to want to focus right in on a point in a topological space . We’re interested in the subsets of in which we could “wiggle around a bit” without leaving the subset. These subsets we’ll call “neighborhoods” of .
More formally, a neighborhood of is a subset of which contains some open subset , which contains the point . Then we can wiggle within the “nearby” set and not leave . Notice here that I’m not requiring itself to be open, though if it is we call it an “open neighborhood” of .
In fact, any open set is an open neighborhood of any of its points , since clearly it contains an open set containing — itself! Similarly, a subset is a neighborhood of any point in its interior. But what about a point not in its interior? If we take a point , but , then is a neighborhood of if and only if there is an open set with . But then since is an open set contained in , we must have , which would put into the interior as well. That is, a set is a neighborhood of exactly those points in its interior. In fact, some authors use this condition to define “interior” rather than the one more connected to orders.
So the only way for a set to be a neighborhood of all its points is for all of its points to be in its interior. That is, . But, dually to the situation for the closure operator, the fixed points of the interior operator are exactly the open sets. And so we conclude that a set is open if and only if it is a neighborhood of all its points — another route to topologies! We say what the neighborhoods of each point are, and then we define an open set as one which is a neighborhood of each of its points.
But now we have to step back a moment. I can’t just toss out any collection of sets and declare them to be the neighborhoods of . There are certain properties that the collection of neighborhoods of a given point must satisfy, and only when we satisfy them will we be able to define a topology in this way. Let’s call something which satisfies these conditions (which we’ll work out) a “neighborhood system” for and write it .
First of all (and almost trivially), each set in must contain . We’re not going to get much of anywhere if we don’t at least require that.
If is a neighborhood of , and , then must be a neighborhood as well since it contains whatever open set satisfies the neighborhood condition . Also, if and are two neighborhoods of then . That is, there must be a neighborhood contained in both and . We can sum up these two conditions by saying that is a “filter” in the partially-ordered set .
So, given a topology on we get a filter for each point. Conversely, if we have such a choice of a filter at each point, we can declare the open sets to be those so that implies that .
Trivially, satisfies this condition, as it doesn’t have any points to be a neighborhood of. The total space satisfies this condition because it’s above everything, so it’s in every filter, and thus is a neighborhood of every point.
Now let’s take two sets and , which are neighborhoods of each of their points, and let’s consider their intersection and a point . Since is in both and , each of them is a neighborhood of , and so since is a filter we see that must be a neighborhood of as well. We can extend this to cover all finite intersections.
On the other hand, let’s consider an arbitrary family of sets, each of which is a neighborhood of each of its points. Now, given any point in the union it must be in at least one of the sets, say . Now tells us first that by assumption, and then that by the filter property of . Thus we can take arbitrary intersections, and so we have a topology.
One caveat here: I might be missing something. Other definitions of a neighborhood system tend to include something along the lines of saying that every neighborhood of a point contains another neighborhood in its interior. I seem to have come up with a topology without using that assumption, but I’m willing to believe that there’s something I’ve missed here. If you see it, go ahead and let me know.
Tonight I’ll be telling the undergrad math club here about a nifty little thing I picked up from Scott Carter last week, who in turn picked it up from John Conway shortly before that: Faulhaber’s Fabulous Formula. The formula itself expresses the sum of the first -th powers as (sort of) an integral:
The question here is what this mysterious is, and the answer to that takes us into the shadowy netherworld of 19th-century analysis: the umbral calculus.
Our mysterious character is defined by a simple equation: . For this is trivially true, as both sides are . For this is clearly nonsense, as it says , or , so let’s just not apply the relation for this value of . For higher , it’s still nonsense, but of a subtler sort. So let’s dive in.
When we consider we get . The terms cancel, and what’s left tells us that . So ? Not quite, because that would make sense!
Let’s look at . Now the relation reads . Again the terms cancel, and now we know that . Thus we can solve to find . Huh?
Moving on to , we find . The terms cancel and we substitute in the values we know for and to find . As I told Scott, “you do realize that this is f___ing nuts, right?”
We can keep going on like this, spinning out values of for all — , , , , , — which have absolutely nothing to do with powers of any base . But we use them as powers in the relation. Remember: this is prima facie nonsense.
But onwards, mathematical soldiers, integrating as to war: we take the integral and break it up into pieces:
Even (most of) my calculus students can take the antiderivatives on the right to get
Now we use the binomial formula to expand each of the terms in the numerator.
and combine to find
But we said that unless ! That is, all these terms cancel except for one, which is just . And so the integral we started with is the sum of the first -th powers, as we wanted to show.
So how do we apply this? Well, we can evaluate the integral more directly rather than breaking it into a big sum.
And now we use the values we cranked out before. Let’s write it out for , which many of you already know the answer for.
What about ? It’s not as well-known, but some of you might recognize it when we write it out.
In fact, given the numbers we cranked out back at the start of the post, we can use this formula to easily write out the sum of the first 9th powers:
all thanks to Faulhaber’s Fabulous Formula.
When we pick a topology on a set , not every subset is open, and not every subset is closed. However, we can still come up with some open and closed sets from any subset .
For the open set, notice that we always have at least one open set inside : the empty set. So we can gather up all the open sets contained in and take their union. Since they’re all contained in the union will be as well. And since arbitrary unions of open sets are still open, it’s an open set. In fact, it’s the largest open set contained in , because it contains all the other open subsets of . We call this the “interior” of , denoted or . Clearly the interior of an open set it the set itself.
Dually, we know that there is at least one closed set containing : the whole space . Then the intersection of all the closed sets containing will be a closed set containing , and will be the smallest such closed set. We call this the “closure” of and write or . As for the interior, the closure of a closed set is the set itself.
Now the complement of the closure of is an open set contained in the complement of . In fact, any other open set contained in the complement of will be contained in this one, so it is the interior of the complement of . Dually, the closure of the complement of is the complement of the interior of .
We can write this fact down categorically as well. Since it reverses subset containment, complementation is a contravariant equivalence from the poset of subsets of (considered as a category) to itself. That is, is equivalent to . The interior and closure operators are covariant functors from to itself, since they preserve containment. The previous paragraph states that these two functors are dual to each other, in the sense that is the same functor as under the above equivalence. So all the really important information is contained in the closure functor.
Now, what do we know about this functor? Well, since is contained in we have a natural transformation . Then since is contained in we have a natural transformation . I haven’t really covered these yet, but it’s straightforward from here to verify that along with these two natural transformations forms a monad. If you’re interested in learning more right away, go check out The Catsters’ series of YouTube videos.
We also can easily check that , and that . That is, the functor preserves all finite coproducts. It turns out that this is enough to characterize the topology in its entirety!
Given a set , a closure operator on is a monad , where is a functor which preserves finite coproducts. This data is equivalent to the four axioms given by Kuratowski:
From here we can define the closed sets of to be those in the image of the functor . From axiom 1 we see that , but this closure must be a subset of , and so is closed. Axiom 4 tells us straight off that is closed. Axiom 3 tells us that the finite union of closed sets is closed. We just need to know that arbitrary intersections of closed sets are again closed.
For this, we note that given any collection of subsets, the intersection lies in each , and so by functoriality for each . Thus we see that . In particular, if all the are closed, then . But since for all , this inclusion is actually an equality, and thus the intersection of the is in the image of the closure functor. And thus we really have constructed the closed sets of a topology on .
Okay, we know what a topological space is. As we might expect, these will be the objects of some category . So we need morphisms to connect them, and for this we will use the concept of a continuous map. In some sense these will be “closeness preserving”, but as we’ll see they work a little differently than the various algebraic categories we’re used to.
Here’s the definition: a continuous map from the topological space to the topological space is a function between the underlying sets so that the preimage of an open set is open. That is, for every open set we can construct , and we require that this set be open in . That is: . We have a special term for isomorphisms in this category: “homeomorphism”. That is, a homeomorphism between topological spaces is a continuous map with a continuous inverse.
This always feels a little weird to me. Over in algebra we take sets, add extra structure, and define the morphisms to be those functions which preserve that structure. Here we’re defining the morphisms to be those which reflect the extra structure. But let’s accept it and move on.
Of course we also want to consider the categorical perspective. What does a continuous map give us in terms of the topology? It’s a functor from the topology on to the topology on , considered as categories! Indeed, if are two open sets in , then — everything that lands in also lands in — and so we send open sets to open sets and inclusion arrows to inclusion arrows. This shows that is a functor.
Even more is true. First, notice that everything in goes to some point of , so . Similarly, nothing lands in the empty subset of , so . Given a family of open sets , a point that lands in their union is in the preimage of at least one of the sets — . Similarly we see that given a finite collection of open sets , a point that lands in their intersection is in the preimage of each of them — . That is, preserves the finite products and arbitrary coproducts that we assume to exist in these categories.
The catch here is that, as far as we’ve come, we can’t really recover a function from this functor . That is, only talks about open sets rather than about points of our spaces. As yet, these two definitions aren’t quite equivalent, and we’re stuck with talking about the map as fundamental and deriving from it the functor .
Okay, so we’ve defined a topology on a set . But we also love categories, so we want to see this in terms of categories. And, indeed, every topology is a category!
First, remember that the collection of subsets of , like the collection of subobjects on an object in any category, is partially ordered by inclusion. And since every partially ordered set is a category, so is the collection of subsets of .
In fact, it’s a lattice, since we can use union and intersection as our join and meet, respectively. When we say that a poset has pairwise least upper bounds it’s the same as saying when we consider it as a category it has finite coproducts, and similarly pairwise greatest lower bounds are the same as finite products. But here we can actually take the union or intersection of any collection of subsets and get a subset, so we have all products and coproducts. In the language of posets, we have a “complete lattice”.
So now we want to talk about topologies. A topology is just a collection of the subsets that’s closed under finite intersections and arbitrary unions. We can use the same order (inclusion of subsets) to make a topology into a partially-ordered set. In the language of posets, the requirements are that we have a sublattice (finite meets and joins, along with the same top and bottom element) with arbitrary meets — the topology contains the least upper bound of any collection of its elements.
And now we translate the partial order language into category theory. A topology is a subcategory of the category of subsets of with finite products and all coproducts. That is, we have an arrow from the object to the object if and only if as subsets of . Given any finite collection of objects we have their product , and given any collection of objects we have their coproduct . In particular we have the empty product — the terminal object — and we have the empty coproduct — the initial object . And all the arrows in our category just tell us how various open sets sit inside other open sets. Neat!
Part of the disappointment I mentioned is that the road I was on just looked so pretty. I’ve said in various places that I agree with (what I understand to be) David Corfield’s view of mathematics as a process of telling good stories, and this was a great story, but unfortunately it just doesn’t quite ring true. Before I purge it, I want to show you the picture of the tensorator as I thought it would work.
Across the top are two tensor products of one span and one object each, and across the bottom are the other two, giving the compositions in both orders. The squares (that look like triangles) at the top and bottom are pullbacks, giving the actual composite spans. Then we can put the tensor product in the middle, and get arrows up and down from the universal properties of the pullback squares. And it even looks like a big tensor product symbol!
I’ve just had a breakthrough today on my project to add structures to 2-categories of spans. I was hoping to generalize from the case of a monoidal structure on the base category that preserved pullbacks. After some discussions with John Baez and Todd Trimble (to whom I’m much indebted), I set off on this new quest and ran into some difficulties. Finally I’ve established that in order to have a well-behaved tensorator we must assume that the monoidal structure preserves pullbacks! This is a bit of a downer, in that I was really hoping to construct a wider class of braided monoidal 2-categories with duals, but at least it covers the cases that originally drew me to the problem. With luck there will still be something interesting here. Anyhow, let’s see how this works.
First we have to consider a span . We want this to be invertible, and further we want its inverse to be its reflection . When we pull back over itself we get a square
Here we can swap and to get a unique arrow with . For this to give us the identity span on we need to have . This tells us that , and we will say for the equal arrows , which is a right inverse for : . Similarly we find a right inverse for .
Now we use the universality of the pullback in the diagram
to give us a unique arrow with , and similarly a unique arrow with . Then we see that , and the universality condition tells us that , while . And thus for the span we started with to have an inverse of the right form we must have .
Now we look for a tensorator . We start with spans
and we must find a span between the pullback objects and . Further, we will want this span to have its own reflection for an inverse, as above. But as we just showed, this means that the two objects at the ends of its legs must always be isomorphic.
Now we can specialize to pick , , , and . Then the one leg of the tensorator span will be , and so the other leg must be as well, no matter what we choose for and ! That is, to have any hope of finding such a well-behaved tensorator, the monoidal product on must preserve pullbacks!
Well, I’m not quite done with the updates of the topics, but I’ve gotten a number of other things done on my break. Now there’s a search bar over on the right, and the WordPress bug for subtopics has been handled. Rather than delay any longer, I guess I should jump back into the thick of it.
Topology is, roughly speaking, the study of spaces where we have an idea of what it means for points to be “close” to each other, and functions which “preserve closeness”. We don’t care about anything but the most general notion of shape. There’s the famous example of a coffee mug and a doughnut being “the same” to a topologist because they both have one hole, and if you make them out of clay you can deform one into the other without making any drastic changes like a sharp cut. In fact, it’s common to say that topology is all about situations like this, where our shapes are made from clay or rubber sheets that can be deformed around, but as we’ll see there are plenty of situations where we can make cuts (as long as we sew them up again nicely) or even weirder things can happen. Deformations are a good intuition for some aspects of topology, but they’re definitely not the most general.
Okay, so how can we get a handle on this notion of “closeness”. The usual way is to take the set of points we’re looking at and define some collection of its subsets as the “open” subsets. Such a collection is required to satisfy a few rules:
- The empty set and the whole set are both in
- The union of any collection of subsets in is again in
- The intersection of any finite collection of subsets in is again in
We call the specified collection a “topology” on the set , and pair of a set and a topology on we call a “topological space. The elements of we call the open sets of , and their complements in we call the closed sets.
Notice here that the collection of closed sets is completely determined by the collection of open sets. This leads to an alternate viewpoint, where we define a collection of subsets of satisfying:
- The empty set and the whole set are both in
- The intersection of any collection of subsets in is again in
- The union of any finite collection of subsets in is again in
Now the elements of are called the closed subsets of the topological space, and their complements are called the open subsets.
We can put more than one topology on the same set , and we can compare different topologies. Let’s say that we have topologies and on a set , so that . That is, every subset of that calls open, does as well. In this case, we say that the topology is “coarser” than , or that is “finer” than . Since we define this relationship by restricting subset containment from to those collections of subsets of which are actually topologies, it defines a partial order on the collection of all topologies on .
The coarsest possible topology is , which says that only the empty subset and the whole set are open. We call this the “trivial” or the “indiscrete” topology on . Conversely, the finest possible topology is , which says that every subset is open. This we call the “discrete” topology on . Useful topologies tend to fall somewhere between these two extremes, but at least we know that has a top and a bottom element for the coarseness relation.
In the middle, let’s say we have some collection of topologies. Then we can define their intersection as subsets of . This will also be a topology, as is easily shown from the definition above. It is the finest topology which is coarser than all the topologies in , and so any subset of has a greatest lower bound.
On the other hand, the union of this collection may not be a topology, which could serve as a least upper bound. However, there is always at least one topology that contains this union — the discrete topology. So we can consider the collection — known to be nonempty — of all topologies which contain the union . The intersection of this collection of topologies will be a topology (as above) which is finer than each topology , and is the coarsest possible such topology. Thus any subset of has a least upper bound.
Together, these results say that the is a complete lattice under the coarseness relation. This turns out to be useful when we have some set we want to put a topology on, and we want to do it in the coarsest possible way subject to a collection of requirements. The fact that is a complete lattice says that we can find the coarsest possible topology satisfying the relations one at a time, and then we can find the coarsest topology finer than each of them.