Interlude -- The Baire Category Theorem
21 Sep 2021 - Tags: analysis-qual-prep
We’ve had a few really detailed posts in a row – summarizing the main theorems of some major objects of study in analysis, and putting these theorems in contexts which (I hope) make them feel coherent and memorable. Today’s post is going to be a bit more relaxed. Two of my best friends (Remy and Alyss) are visiting me, so I want a slightly shorter form post to make sure I have time to spend with them. The Baire Category Theorem seems like the perfect topic!
(Future Chris here, they left on Sunday, but I’m only now getting this post up. My qual is tomorrow morning, and I ended up studying for that instead of revising this. I’m still going to write up how I organized my thoughts on Fourier Analysis, but it’ll probably be after I actually take the exam. Wish me luck!)
First, let’s remind ourselves of the (deceptively simple) statement of the
Baire Category Theorem. Recall a set is called
Meagre if it is a countable union of nowhere
dense sets. This forms a
Let
Then
Another way of phrasing the result, which sounds less obvious by expanding some of the complexity trapped in the word “meagre”, is very useful in practice:
Let
Then
This result, which especially in its first form seems obvious, has a surprising number of consequences. I’ll give some applications here, particularly applications which are relevant for qual-related counterexamples, but if you want more you should see here for a long list, and here for what is essentially a “user manual” for proving existence results using the Baire Category Theorem!
The idea behind proving theorems with the BCT is to write think about
what properties you want some object to satisfy. Ideally there are only
countably many such properties
If you find yourself in this scenario, then the BCT promises you can find an element which satisfies all the properties simultaneously!
Contrapositively, if we have countably many meagre conditions, then we can avoid all of them simultaneously!
Let’s see a few examples in action:
Most continuous functions
Here “most” means “a dense
Now we want to characterize those
In full, we want to look at the sets
As a quick exercise, you should verify that if
A (slightly tedious) argument shows that each
This tells us that most continuous functions are not differentiable anywhere.
This was supposed to be a shorter, lower effort post, and I definitely
succeeded in that regard, haha. I was originally going to also prove that
if
I know there is a BCT argument, but I’m feeling too lazy to come up with it myself, and all the places I checked proved this by using the uniform boundedness principle5 (which does indirectly use BCT, but I would like to be a bit more explicit in this post).
If someone happens to have an argument on hand, I’d love to hear about it! Until then, the qual is tomorrow, so wish me luck! Next time will be about fourier analysis, and it’ll be the last post in the series. See you there ^_^.
-
Though which sets are classified as “small” can differ! There are subsets of the unit interval which have lebesgue measure
despite being meagre.The precise differences between the
-ideals of meagre and nullsets is a topic of some classical interest in descriptive set theory. If we assume , then there is an involution on which swaps meagre and null sets. See here, for instance. ↩ -
Or a locally compact hausdorff space ↩
-
Again, or a locally compact hausdorff space ↩
-
If you’ve never seen this proven, it’s worth doing it yourself at least once! It’s not hard, and will help you gain some comfort with the definition of meagre. ↩