January 5, 2012
During one of my previous qualifying exams, one of the questions looked, at first, to be a basic Fourier Analysis question. The hint, though, was to use the Baire Category Theorem. The problem and hint looked so unrelated that the phrase has become somewhat of a running joke for some of us in the department (“How are we going to prove this is a cofibration?” “Oh, easy, just use the Baire Category theorem.”).
The more I read about it, the more I realized that the Baire Category theorem pops up much more than I realized and in a number of surprising and diverse topics. This post is just to introduce you to the BCT and to show off some of its raw power. One of the results below is one that I genuinely don’t believe — that is, I know it has to be true (it was proved!) but it just seems so outrageous that I can’t being myself to believe it! But, we’ll get to that.
Leading up to the Theorem.
There are a few different formulations of the theorem, so I’ll pick the one that turns out to be most common in what I’ve read. First, let’s quickly define some terms.
Definition. A complete metric space is a metric space where each Cauchy sequence converges to a point in the space.
For example, is complete, but is not complete since in the latter space we can take a sequence like: 1, 1.4, 1.41, 1.414, … which are all rational numbers that converge to , but isn’t in the rational numbers.
Definition. A subset of a space is dense if the closure of the subset is the entire space.
For example, is dense in since the closure of is . The irrationals are also dense in . The integers, , are not dense in because the closure of the integers in is just (why?).
Q: What are the dense subsets of a non-empty discrete space? Does this make sense “visually”?
Definition. The interior of a subset is the largest open set contained in . In metric spaces, it’s the union of all the balls contained in .
The next definition might be a little strange if you haven’t seen it before and, in fact, there are two nice (equivalent) ways of thinking about this concept. I’ll list both.
Definition. A set is nowhere dense in some space if either (and hence both):
- The interior of the closure of in has empty interior.
- ; that is, the closure of the compliment of is dense in .
This last definition might seem a bit strange to you, but the “image” of this is the following: a nowhere dense set is literally a set which is not dense anywhere — but what does that mean? Density isn’t usually considered a local property. How do we usually check density, though? We take the closure of the set and if it’s the whole space then we say the set is dense. If we wanted to make this more “local”, what we could say is that we don’t get any neighborhoods of the whole space. This is the same as saying the closure of our subset has no open neighborhoods inside of it or that it has empty interior. This givers us the first definition. The second is a direct consequence which is sometimes more useful to think about depending on your original subset.
Example. We have with the standard topology. Is nowhere dense in ? Note that and this has empty interior. By the first definition, must be nowhere dense in . If we wanted to use the second definition, we could say that the compliment of in is just everything but the integers; but the closure of this is which means the compliment of is dense in giving us the same conclusion. This is not surprising, since the elements of are “pretty far apart” in .
Example. Our favorite limit-point example, is a subset of . Is it nowhere dense? Using the first definition, the closure is just which has empty interior (why?). With the second definition, the compliment of has closure equal to and so it is dense. In either case, this proves that is nowhere dense in the reals.
Example. The ball . The closure of this set is which has interior equal to . Therefore, this set is not nowhere dense. That’s a bit sad. But you can see that this set really is “somewhere dense” visually.
Example. The Cantor set is closed (there are a number of cute ways to prove this) so to show it is nowhere dense in it suffices to show that it has empty interior by the first definition. Prove it!
You may have been surprised with the last example, or even the example. For the latter one, it seems like the points are really “getting close” to 0, so maybe some density should occur; it’s not the case, though! For the Cantor set, many points in this set can be “very close” to one-another so it is a bit surprising that this set is not dense anywhere. If one goes through the construction slowly, you’ll see why the points are “just far enough apart” to reasonably be considered (that is, intuitively considered) nowhere dense.
Why would we ever care about nowhere dense sets? Well, they’re kind of crappy in the following sense: even if we unioned a countable number of nowhere dense subsets together, we could never recover the original set. An analogy from anthropology would go like this: we might be able to find tons of copies of some ancient book, but if each of them are missing a whole lot (nowhere dense) then even combining them all together we can’t recover the original manuscript.
Oh. And why do jigsaw puzzle makers make each of the puzzle pieces look like that?
Because if they made them nowhere dense, we would never be able to finish the puzzle. Buh-dum-tsh.
Theorem (Baire Category Theorem). A non-empty complete metric space is not the countable union of nowhere dense sets.
This shouldn’t be too scary to you at this point. Let’s do some immediate consequence examples.
Example. Do you think we could union a countable number of things that look like together and get something like the real numbers? That is, maybe we could take
and so on, and union them all together to make .
Nope. The BCT says to not even try. Each of these translates of is nowhere dense, so a countable union of them would not get you .
Non-Example. What if we did the same thing as above, but we wanted to union them all together to get , the rationals. What would Baire say about that? Probably, “Don’t use my theorem with this, because isn’t a complete metric space.”
Non-Example. What if we tried the same thing with the translates above as subsets of , but we union them all and then union that with the irrational numbers. In this case, too, we can’t use Baire, because the irrational numbers are actually dense in the reals; the BCT doesn’t say anything about unions with dense subsets.
Example. What if we have a complete metric space with no isolated points? Can it be countable?
(Recall, an isolated point is one which has a neighborhood not containing any other point of the space. For example, if we took in the subspace topology of the reals, then is an isolated point of this subspace since it, itself, is open and it contains no other point of the subspace.)
In this case, we can cleverly use the BCT to show that every complete metric space with no isolated points is uncountable. Suppose it were countable. Then every point of the space is nowhere dense (! Why is this the case? What is the closure of a point? What is its interior? Does this argument show why we cannot have isolated points?). But since our space is countable, it is the countable union of its points, each of which is nowhere dense. Baire would have a real problem with that. This contradicts BCT, so such a space must be uncountable.
Why is it called the “Category” theorem?
This “category” isn’t the same category as in category theory. The term comes from the original language that the theorem was stated in. It’s a bit ugly, so I’ve yet to mention it, but I might as well spell it out for you now:
- We call a set of First Category if it can be written as a countable union of nowhere dense sets. This is also called a meagre set. (Amusingly, the meagre is also the name of a fish.) Sometimes this is Americanized (that is, bastardized) and pronounced as “meager set” which would mean something like, “thin set” or “set having little substance” which is, intuitively, what this set should “look” like.
- We call a set of Second Category if it is not of first category.
We also have what’s called a Baire Space, which is just a space where every non-empty open set is of second category. This means, intuitively, that an open set needs to have some “density” to it. The original Baire Category theorem stated that every complete metric space is a Baire Space. This is “almost” equivalent to what we have above; see below.
Other Formulations of the BCT.
There are three distinct formulations of the BCT that are generally given (the texts I have in my office give at most two, but the third is a consequence of the first so this is perhaps why). Right above this, we give one of them.
BCT (1). Every complete metric space is a Baire space; that is, every open subset in a complete metric space is not the countable union of nowhere dense sets.
BCT (3). Every complete metric space is not the union of nowhere dense sets.
I list these together because it’s clear that the third is a consequence of the first. (These numbers are because the second BCT is actually called “the second BCT” in a few places.)
BCT (2). Every locally compact Hausdorff space is a Baire space. Recall that locally compact means that every point of our space has a compact neighborhood.
Note that it’s not the case that the first BCT implies the second, and it’s not the case that the second implies the first. It would be nice if there were some meta-BCT that implied both of them, but when we write down exactly what we need for something to be a Baire space, we’re left with a couple of choices; if you’re interested in this, you should check out the proofs for both of these theorems and see where we’re using locally compact verses where we’re using complete.
A Surprising Result.
One of the coolest results I’ve seen using the BCT (and, indeed, one of the coolest results I’ve seen perhaps in general) is the following.
Maybe, first, it’s nice to ask the following question (that you can even ask to your friends!): what’s your idea of a “general” continuous function? That is, just think of the most generic continuous function you can. Visualize it. Maybe draw it. What do you have? Probably some sort of polynomial. (When I do this, I actually draw fifth degree polynomials most of the time.) Is this really so generic?
First, let’s just define exactly what we mean by “generic.”
Definition. If is a subset of some space , the compliment of is nowhere dense, and all points in share some property, then we call this property generic of .
This should be reasonable: something genertic should be representative of the whole. That is, if something is not generic, then it should not happen very often; in our language, the set of non-generic things should be nowhere dense.
Theorem (Banach, Mazurkiewicz, 1931). The generic (in the sense of the above definition) continuous function on a compact interval (that is, )) is nowhere differentiable, and not monotone on any subinterval of .
Just think about that for a second. Let it sink in. Nowhere differentiable is strange enough; we can (and will) use the BCT to talk about a nowhere differentiable function. But, what’s more, it’s not monotone on any subinterval, no matter how small. And, yes, this function is generic. What.
As a note, somewhat strangely, because of the above definition of “generic” it is entirely possible to have two different kinds of “generic” things representing some space. The definition doesn’t say anything about the generic property being unique and, in general, it isn’t.
BCT in Functional Analysis.
Whenever I talk to students who have had basic point-set topology, I’m not surprised if they haven’t seen the BCT; it’s near the end of the point-set part of Munkres, and one natural stopping point for a semester-long point-set class is Urysohn’s lemma which is done before BCT. I don’t remember learning the BCT in point-set when I took it and it wasn’t until Functional Analysis that I had to study it again. This is a shame for a few reasons, but I wanted to point out just two.
- This theorem is able to be proved (or at least stated) after complete metric spaces are defined and would take only around one lecture to do with applications. Depending on the speed of the professor and the way one works through the book, one probably gets to metric spaces around the middle of the semester. The students have had analysis, and are, therefore, familiar with completeness (or, at worst, they’re comfortable with Cauchy sequences!) so it is not difficult to define complete metric spaces. What else would we need? Nowhere-dense sets. That’s all.
- This can be a great “visual” theorem in some spaces. One common problem point-set topology students seem to have is that they will work through definitions without developing any kind of intuition behind what something is (eg, limit points). Therefore, it’s nice to learn properties of spaces we know and love. The BCT is easily applied to the reals, the irrationals, the rationals, and the cantor set. It also is a great brain-stretching exercise to think about how “close” points can be in a nowhere dense set on the real line.
This may be a non-issue for those of you whose point-set class included this, but I wanted to point is out nonetheless. I would go as far as saying that this is “more important” to cover in a first course in point-set than the proof of Urysohn’s lemma [which is the first “deep” result in Munkres]. Of course, the proof is important, but in an introductory course I feel that proving Urysohn’s lemma for metric spaces and stating it for general spaces is a better way to go. Feel free to disagree and yell at me!
Edit: This rant used to be part of the BCT post I made recently, but because it’s not really mathematics and more about teaching I decided to post it on its own so I don’t have to subject everyone to reading my rants.
December 20, 2011
I’m going through a few books so that I can start doing lots and lots of problems to prepare for my quals. I’ll be posting some of the “cuter” problems.
Here’s one that, on the surface, looks strange. But ultimately, the solution is straightforward.
Problem. Find an uncountable subset such that has empty interior.
November 20, 2011
After proving this "Deep Result" [cf. Munkres] a professor will (hopefully) say something like: "Yes, the proof is important, but what does this theorem mean? And what does it mean for spaces which are sufficiently nice, like metric spaces?"
Let’s state the result just so we’re all on the same page.
Theorem. The topological space is normal (that is, every pair of disjoint closed subsets can separated by disjoint neighborhoods) if and only if any two disjoint closed sets can be separated by a function.
The proof of this is not obvious. In fact, it is quite involved. But if happens to be a metric space, we can make the proof significantly easier by using a function that is similar to a distance function with a few minor modifications. I call this function the standard Urysohn function for metric spaces for lack of a better (or shorter) name. The function is as follows:
but before proving this theorem in the metric space setting, let’s look at this function. If are disjoint, then it is clear that the denominator is non-zero (why?) so this is defined everywhere. If we have that this function evaluates to 0. If then we have that this function evaluates to 1. And the function (since the distance is always non-negative) achieves values in for every point not in or (convince yourself that this function is continuous and only achieves the values 0 and 1 if the point is in either or respectively). This function, therefore, separates and .
The next thing you should think about is: what do the preimages of look like? Draw some pictures! Here’s some pictures to start you off to get the idea. The first is just in and the second is in the real plane. Both have the standard topology.
This one is potentially not to scale, but you’ll see that we essentially have a linear relation (since the sum in the denominator will always be 1 in this case) except when we start looking at points that are less than all the points in or greater than all the points in . What happens in those cases?
I’ve left this "face" as an exercise because when I did it I was kind of excited about the result. What do the preimages look like? Does this look like something you’ve seen before; maybe in physics? Could you modify the Urysohn equation above to make it seem MORE like something in physics?
So now that you’ve seen these pre-images, it should be relatively clear how to create neighborhoods around each of the open sets. It still takes a bit of proof, but it’s nowhere near the difficulty of the standard Urysohn’s.
I encourage you, further, to draw some pictures. My rule of thumb is: draw at least five different pictures; four easy ones and a hard one. Find out where the preimages are in each of these. Remember, too, that not every metric space looks the same; what would this look like, for example, in the taxi-cab space? Or in some product space? Or in the discrete space…?
October 16, 2011
In the last post, I posted a topology on where the open sets were
You can check that this is, in fact, a topology and it, in fact, has exactly elements. But this is also a really nice topology for intro-topology students. Why? Because it has some strange properties which are easy to get your hands on and prove! Let’s show some of these. For the sake of brevity, let’s call this topology .
is not Hausdorff.
This is easy to see. Pick the elements 1 and 2. The only set which contains 1 but not 2 is , but there is no set which contains 2 but not 1.
Two points are topologically indistinguishable if they have exactly the same neighborhoods; a space is one where each point is topologically distinguishable. It is trivial to check that this holds.
is connected. (!)
This is somewhat surprising to me (especially because of how disjoint it "seems"!) but if we attempt to find a separation we find that it is impossible: every open set contains 1 except the empty set, so, in particular, no two open sets have trivial intersection unless one is empty.
A function is continuous if and only if the sets of all which map to are open (in the standard topology on .
This is simply taking preimages. I mention it because actually DRAWING the function on the "plane" (where the "x-axis" is and the "y-axis" is we get something which does not at all look continuous (unless it is constant —). This is perhaps a good exercise for intro topology students to work out so that they can see that not all continuous maps "look" continuous. Of course, there are many other ways to see this using other spaces.
Limits? What about the sequences in this set? Let’s remind ourselves what convergence is: if for every open set containing there is some such that for all we have . Note then that:
converges to .
converges to .
and so on. But then we have
which converges to 2, but not to 1 (why?).
which converges to 3, but not to 2 or 1 (why?).
and so forth. In fact, if we have an infinite number of the numbers then this sequence will only converge to which is kind of a neat characterization. In fact, this takes care of most (all?) of the sequences.
Limit Points? A limit point of a subset is a point such that for every open containing we have .
Consider the subset and let’s see if there are any limit points. What about 2? Well, we have that so the intersection of this with is nontrivial. Similarly, the other open sets containing 2 have nontrivial intersection with which includes more than the element 2. What about 3? Same deal. And 4? Yes. So we can now ask: what’s the closure of ?
Consider and we will start looking at the element 3. Is 3 a limit point? For the same reasons above, it is. So what is the closure of this set?
Let’s be a little adventurous and ask what about . For the same reasons as above, are all limit points, but what about 2? In fact, it is.
Last, (and this will complete our "characterization" of the closures of our subsets) what about . Clearly we will have as limit points, but the odd one out is 1. And 1 is not a limit point (why not?). That’s a bit sad. So, taking the least element in the subset allows us to describe the closure: it’s the set of every element greater than or equal to it.
Closed sets. It’s a bit interesting to think up what sets we got from taking the closures of each subset, and then to just start with our open sets and take compliments. Do we get the same thing? Neato.
Compactness? A bit trivial in this case. Why? (Maybe extending to infinite things would make this cooler…)
Here’s something weird, too: what is a path in this space? What is a loop? How could we construct something like the fundamental group for this? Homology groups? To those with more experience, these questions might be easy; but it is not obvious to me (at least at this point!) how these constructions should go.
October 14, 2011
The following question came up today after a brief and somewhat unsatisfying talk I gave on the intuition behind topologies:
Question: Given , can we construct a topology such that has exactly elements?
The solution wasn’t obvious to me until I looked at the pattern of numbers on the back of a thing of Dried Papaya Chunks. For we can simply take and to be our sets, but for we need to be a bit more clever. But not much more. Consider:
This is a topology on with exactly elements.
In these examples, the elements which are nested inside each other and is somewhat reminiscent of the "particular point" topology — the topology in which a set is open if and only if it contains some predetermined point (or is empty). In the finite case, how many elements are in the particular point topology? Let’s do a few examples.
If our universe is , there are two open sets in the particular point topology (empty set and the whole set).
If our universe is , then we have three open sets in the particular point topology.
If our universe is , then we have five open sets in the particular point topology.
If our universe is , then we have nine open sets in the particular point topology.
If our universe is , then we have seventeen open sets in the particular point topology.
If we exclude the empty set, it seems like we will have elements in our particular point topology. Why is this?
(Hint: Take the last example above with seventeen open sets and suppose the element 5 is your particular point. Write down all the open sets containing 5. Now take your pen and scribble out the 5’s in each of those open sets — do the resulting sets look familiar? Hm.)
Neat. If you take other finite topologies, are there any neat patterns that arise when you increase the number of digits? Is it possible to associate some sort of polynomial or invariant with finite topologies? With infinite topologies? Would this even be useful?
[Big Edit: Thank you to the commenters below (and especially the latest one) who pointed out some of my previous statements only held true in a specific case. I’ve streamlined the proof, and I think everything is kosher now. I’ve also added some counterexamples to the end.]
I was reading through some of the earlier chapters of Bredon and I got to one of those questions that should be easy but I couldn’t recall the argument off of the top of my head. The actual question is slightly more general, but the question we’ll answer first is:
Problem: If is connected, show is connected.
There are several nice definitions of connected to choose from. In particular, we have:
A subset is connected if one (and hence all) of the following hold:
- There DO NOT exist disjoint non-empty closed (with respect to ) sets such that and but .
- The only subsets of which are both open and closed in the relative topology are and .
[Note: In a former draft of this post, I had (incorrectly!) stated that a subset is connected if we did not have two disjoint open sets which do not intersect trivially, but which union to . This can be shown to be false (which was done by a clever commenter below), but it will work if (and therefore in the subspace topology) is a metric space.]
To solve this, we will use the first statement above but, for fun, try to prove it yourself using the second statement. We’ll be using the contrapositive, meaning we will prove if is not connected, then is not connected. This is equivalent to proving the theorem above (why?).
Solution. Let it be the case that is not connected, so that there exists disjoint closed sets neither of which intersect trivially. Suppose that is connected; then for any two disjoint closed sets with we have that one of these closed sets intersects trivially (or else it would not be connected). In other words, wlog we have . Taking the closure of both sides, we have , but since and are disjoint this implies that , contradicting the fact that neither nor may intersect nontrivially.
In fact, nearly the same kind of argument shows that if there is some such that with connected, then is also connected. This is pretty nice; it tells us we can add limit points to our connected space (even if we only add some limit points and not others!) and it remains connected.
Some counter examples…
One might ask, "Hey, what if is connected? Then is connected?" And a minute of thinking should provide a counter-example. If not, then just think about and its closure.
Additionally, one might think that if a set is connected, then its interior (if it is nonempty) would be connected. This is true in the case of the closed disk , but if we were to put two disjoint disks in the plane and connect them together with a line (this will look like a barbell, or a pair of eyeglasses) the interiors will be two disjoint open disks, which are clearly not connected.
Maybe, though, the boundary of a connected set is always connected (if it is nonempty). To do away with this idea, just think about the interval in .
A question to think about (assume all of these sets are non-empty):
- If the boundary of some set is connected, is connected? (For this one, think about a figure-8 in the plane.)
- If the interior of some set is connected, is connected? (For this one, think about what the interior of a point is, and then let be some nice set with non-empty interior union some point.)
- If the interior and boundary of some set are connected, is ? (For this one, you might have to think a bit. I’ll try to hide an answer a little bit so you can work on it.
Consider the set of positive rational numbers, union the interval . The interior will be which is connected, and the boundary will be which is connected, but the set itself is not connected (for the same reason the rationals are not connected).
Every so often I like to go back and read through some basic algebra or basic point-set topology and try to think of new ways to look at the easier, more fundamental problems. I’ve been going through Bredon’s Algebraic Topology to prepare for a topology topics class next semester and there are two wonderful things about this book:
- It is concise*. When you read Bredon and Hatcher back-to-back you’ll understand what I mean here: while I do love Hatcher’s long motivating paragraphs (and, in fact, I attempted to mimic his method of explaining things in depth when I first began this blog), I tend to get lost in his prose at times. (On the other hand, Bredon’s book had significantly less motivation. Significantly less.)
- The questions are few, but proud. No one can deny that Hatcher’s book has a ton of enlightening and challenging questions. On the other hand, Bredon’s questions are generally more focused toward the section immediately preceding it (helping the beginning student) and are generally quite interesting (which is not to say the Hatcher ones are not!). The nice part about Bredon’s book is that you could easily assign students a section and ALL of the associated questions and they would probably not complain too much since there’s usually only three or four associated questions. Not so with Hatcher.
(*Perhaps not as concise as May’s book, though!)
March 22, 2011
What’s the weakest separation we can have in a topological space?
Well, “no separation” is pretty weak. But this creates the trivial topology and that’s a bit boring. So let’s say this:
Definition: A topological space is or Kolmogorov if for every two points we have that there exists a neighborhood such that either and or and .
In other words, a space is if for every pair of points there is at least one open set which contains one and doesn’t contain the other. This is a pretty weak separation condition. Certainly, every Hausdorff space is , but there are ones which are even weaker which satisfy this condition. Let’s try to construct a really easy one.
March 12, 2011
I cannot find a suitable picture of a linked double torus, which makes me think it’s called something else. Luckily, because it’s on the cover of Armstrong’s book (and I don’t think Springer would mind some free advertising) we can reproduce it here:
Now, how can we take this "linked" double torus and make it an "unlinked" double torus without cutting anything up? This one kept me up last night, but the solution turns out to be quite easy once you see it.
If you don’t want the full solution, a hint might be: start thinking of the double torus as two handles on a sphere. What can you do with these handles?
The picture solution after the jump.