Classic applications of Baire category theorem

Question

I've seen Baire category theorem used to prove existence of objects with certain properties. But it seems there is another class of interesting applications of Baire category theorem that I have yet to see.

First I found this MathOverflow problem:

Let $f$ be an infinitely differentiable function on $[0,1]$ and suppose that for each $x \in [0,1]$ there is an integer $n \in \mathbb{N}$ such that $f^{(n)}(x)=0$. Then $f$ coincide on $[0,1]$ with some polynomial.

I found another one from Ben Green's notes:

Suppose that $f:\mathbb{R}^+\to\mathbb{R}^+$ is a continuous function with the following property: for all $x\in\mathbb{R}^+$, the sequence $f(x),f(2x),f(3x),\ldots$ tends to $0$. Prove that $\lim_{t\to\infty}f(t)=0$.

Are there any other classic problems of this type?

Answer 1

Many applications of Baire Category Theorem, such as the characterization of real polynomials you cited, follow from the strengthening of a pointwise statement
$$ \forall x \in X, \exists d \in D, \ P_d(x)$$ to an uniform statement $$ \exists d \in D, \forall x \in X, \ P_d(x), $$ where $D$ is a enumerable set and $X$ is a complete metric space.

For this, one uses the following strategy (translated from BwataBaire, a wiki in French that contains a lot of applications of Baire Category Theorem):

1. Define the set $F_d = \{x \in X \mid P_d(x) \}$.
   Since the pointwise statement is valid, we have that $ X = \bigcup_{d \in D} F_d $.
2. If we are lucky, the set $F_d$ is closed in $X$ for each $d$. By Baire Category Theorem, then there exists a $D$ such that $F_D$ has a non-empty interior.
3. By using that $\mathrm{int}(F_D)\ne\emptyset$, we try to prove that $F_D = X$, from which will follow the uniform statement.

As an example, you can try to use this strategy to prove the following:

Let $X$ be a Banach space and $f:X\rightarrow X$ a continuous linear application such that $$ \forall x \in X, \exists n \in \mathbb{N},\ f^{n}(x)=0. $$ Then $f$ is nilpotent, that is, $$ \exists n \in \mathbb{N}, \forall x \in X,\ f^n(x)=0.$$

It is interesting to remark that there also exists a lot of another common strategies to change a pointwise statement to an uniform statement. One good place to learn about them is this article from tricki.

P.S. There is also a community wiki question in math.stackexchange with a lot of this kind of applications of Baire Category Theorem: "Your favorite application of the Baire category theorem".

Answer 2

I have multiple reasons to single out one particular application of Baire Category Theorem. First, it is a result about holomorphic functions, so it belongs outside the-usual-suspects of real analysis, topology or functional analysis. Second, it is due to William Fogg Osgood, who formulated a version of Baire's theorem before a general version of Baire's theorem was available. Let $D$ be a domain (in the complex plane) and let $\{f_n\}$ be a sequence of functions analytic in $D$. Suppose $f_n(z) \to f(z)$ for each $z \in D$. Then $f$ is analytic in an open set $D_1 \subset D$ which is dense in $D$, and convergence is uniform on compact subsets of $D_1$.

Originally it appeared in MR1502274 Osgood, W. F. Note on the functions defined by infinite series whose terms are analytic functions of a complex variable; with corresponding theorems for definite integrals. Ann. of Math. (2) 3 (1901/02), no. 1-4, 25–34.

It is stated and proved (as Theorem 2.1.25) in the book in Spanish by Brito mentioned in Ljubomir Cukic's answer, and in the article

MR0328028 Zalcman, Lawrence Real proofs of complex theorems (and vice versa). Amer. Math. Monthly 81 (1974), 115–137. (Reviewer: D. Gaier)

These publications refer to Osgood, unlike the article MR2463290
Krantz, Steven G. Complex analysis as catalyst. Amer. Math. Monthly 115 (2008), no. 9, 775–794, which also states and proves this theorem.

Answer 3

There are a very important applications of the Baire Category Theorem: the Banach-Steinhauss Theorem, the Vitali-Hahn-Saks Theorem, the Nikodym Theorem,...There is a nice book on the Baire Category Theorem (Spanish, 564 pages): Wilman Brito, El Teorema de Categoría de Baire y aplicaciones,

http://www.ciens.ula.ve/matematica/publicaciones/libros/por_profesor/wilman_brito/Teorema_de_Baire_Aplicaciones.pdf

http://www.amazon.com/El-Teorema-Categor%C3%ADa-Baire-aplicaciones/dp/3846575909

Answer 4

Yes, there are many applications besides proving the existence of objects with pathological properties. I recommend the nice book MR0584443 Oxtoby, John C. Measure and category. A survey of the analogies between topological and measure spaces.

Answer 5

I got some mileage out of the Baire category theorem in abelian group theory, in particular in the study of the group $\mathbb Z^{\aleph_0}$, the additive group of sequences of integers (which, contrary to some people's expectations, is not free). See the papers "Baer meets Baire" (joint with John Irwin) at http://www.math.lsa.umich.edu/~ablass/bb.pdf and "Specker's theorem for Nöbeling's group" at http://www.math.lsa.umich.edu/~ablass/spno.pdf .

Answer 6

As mentioned above, a classical use of the BCT in functional analysis is to prove the three central results around the closed graph theoorem. Less well known and considerably more subtle is a method used by Saks to prove what is now known as the Vitali-Hahn-Saks theorem on the $\sigma$-additivity of limits of sequences of measures ("On some functionals", TAMS 35 (1933), 549-555). The novelty of this example lies in the fact that he applied the BCT to the unit ball of $L^\infty$, regarded as a metric space with the $L^1$ norm. The fact that this is not a vector space makes the proof more delicate.

Answer 7

Here is a cute application of Baire category theorem that is in the spirit of your examples.

Assume that $f: {\mathbb{R}}^{\mathbb{N}} \rightarrow \mathbb{R}$ is a Borel measurable function with the property that if $x =^+ y$, then $f(x)=f(y)$, where $x =^+ y$ if and only if $\{x_n: n \in \mathbb{N}\}=\{y_n: n \in \mathbb{N}\}$ for $x,y \in {\mathbb{R}}^{\mathbb{N}}$. Then, there exists $x \in {\mathbb{R}}^{\mathbb{N}}$ such that $f(x)=x_k$ for some $k \in \mathbb{N}$.

We can consider Cantor's diagonalization argument as a Borel map taking a sequence of reals and producing a real different than any element in the sequence. This theorem tells you that there is no Borel way to do diagonalization in a "uniform" way. "Uniform" here means that the sequences consisting of the same elements are diagonalized with the same element.

This is the most basic version of Friedman's Borel diagonalization theorem. In On the necessary use of abstract set theory, Advances in Mathematics, 41 (1981), 209-280, Harvey Friedman proves this result (Proposition C, p. 229) using a forcing argument. Though, in the appendix of the same paper, he gives another proof based on the Baire category theorem. An unusual feature of this proof is that you pass to a different space and apply Baire category to ${\mathbb{R}}^{\mathbb{N}}$ where $\mathbb{R}$ is endowed with the discrete topology.

Answer 8

Another application in number theory: The maximal unramified extension $K^{nr}$ and the algebraic closure $\bar{K}$ of a $p$-adic local field $K$ are not complete.

Answer 9

I proved in 1961 (Marcinkiewicz's student competition):

THEOREM There does not exist any $\sigma$-compact metric space $(U\ d)$ such that every $0$-dimensional metric compact space of diameter $\le 1$ can be embedded isometrically into $(U\ d)$.

My original proof was quite heavy. Later Aleksander Pełczyński provided a new and simpler proof which used Baire category.

Answer 10

The famous Ornstein Isomorphism theorem in ergodic theory (any two Bernoulli shifts with the same entropy are isomorphic) has a simplified proof due to Burton and Rothstein, which uses Baire Category.

Answer 11

The Uniform Boundedness Principle, Open Mapping Theorem, Closed Graph Theorem are all derived via Baire Category theorem (they can all be derived from each-other as well).

Answer 12

One can prove--using the Baire category property--that certain Hausdorff spaces (e.g. $\mathbb Q$) do not majorize any Hausdorff compact space, or even H-minimal (or even certain more general spaces).

Answer 13

Here's a good one, which essentially shows that every closed $2$-dimensional subset of the plane contains a region of the plane.

Theorem. Let $X\subset \mathbb R^2$ be closed and $x\in X$. Suppose that for every $\varepsilon\in (0,1]$, $X$ contains an arc along the circle of radius $\varepsilon$ centered at $x$. Then $X$ contains a whole region of the plane.

Proof: Enumerate $\mathbb Q\cap [0,2\pi]$ as $q_1,q_2,q_3\ldots$. For each $\varepsilon\in (0,1]$ let $S_\varepsilon(x)$ be the circle of radius $\varepsilon$ centered at $x$. When $n\neq m$, let $S^{n,m}_\varepsilon(x)$ be the arc of points of $S_\varepsilon(x)$ at angles between $q_n$ and $q_m$ (in radians). Define $$A_{n,m}=\{\varepsilon\in [0,1]:S^{n,m}_\varepsilon(x)\subset X\}.$$ By assumption, $$X=\bigcup_{n,m\geq 1}A_{n,m}.$$ Note that each $A_{n,m}$ is closed in $(0,1]$, since $X$ is closed in $\mathbb R^2$. By BCT, there exist $n$ and $m$ such that $A_{n,m}$ contains an interval $I$ of $(0,1]$. Then $X$ contains the region consisting of all points $z$ such that the angle of $z$ relative to $x$ is between $q_n$ and $q_m$, and $d(x,z)\in I$.