Baire category theorem

Contents

Baire category theorem is a fundamental result in Functional Analysis about the characterization of complete metric spaces with various applications.

Before the formulation of Baire's category theorem we need the two definitions.

Definition (Dense set)
A set

A

is called dense in a metric space if the closure of

A

is the whole space.

Definition (Nowhere dense set)
A set

E

is called nowhere dense in a metric space

X

if every nonempty open set in

X

contains a ball of a positive radius without points of

E

Theorem (Baire's category theorem)
If

X

is a non-empty complete metric space such that

\begin{equation*}X = \bigcup_{n=1}^{\infty} C_n ,\end{equation*}

where the sets

C_n

are closed. Then at least one of them contains an open ball of positive radius.
If

\begin{equation*}X = \bigcup_{n=1}^{\infty} A_n ,\end{equation*}

where

A_n

are arbitrary sets, then at least one of

A_n

is dense in some ball of a nonzero radius, i.e. a complete metric space cannot be the countable union of nowhere dense sets.

History and Motivation

In the late nineteenth century, Baire introduced in his doctoral dissertation a notion of size for subsets of the real line which has since provided many fascinating results. In fact, his careful study of functions led him to the definition of the first and second category of sets. Roughly speaking, sets of the first category are 'small,' while sets of the second category are 'large.'

The video serves only as an illustration. However, the visualization of first category sets reminds the integer lattice, which is actually a first category set

The word “category” in the name of Baire’s theorem is explained by the following terminology: sets that are countable unions of nowhere dense sets are called first category sets and all other sets are called second category sets. Baire’s theorem asserts that a complete metric space is not a first category set.

Dense Sets on $\mathbb{R}$

1 . Rational numbers

\mathbb{Q}

Between any two real numbers, you can always find a rational numberFor example, between 1 and 2, you have

\frac{3}{2}, \frac{4}{3}, \frac{5}{3}

, etc.

2. Irrational numbers

\mathbb{R} \setminus \mathbb{Q}

Similar to rationals, between any two real numbers there's always an irrational number. For example,

\sqrt{2}

\pi

e

are irrational, and you can add any rational to them to get more irrationals

3. The set

{p + q\sqrt{2} : p,q \in \mathbb{Q}}

This is dense because such numbers can approximate any real number arbitrarily closely

Nowhere Dense Sets on $\mathbb{R}$

1. Any finite set

{a_1, a_2, ..., a_n}

For any point in this set, there's an open interval around it containing no other points from the set

2. The set

\mathbb{Z}

of integers
Between any two integers, there's an open interval containing no integers.For example,

(1.5, 1.9)

contains no integers.

Proof of Baire's category theorem

The foolowing proof is taken from [Bogachev] pages 15

Proof of Baire's category theorem

The foolowing proof is taken from [Bogachev] pages 15

\quad

Proof. We can assume that the first set

C_1 \neq \emptyset

since they cannot all be empty and dropping any empty sets does no harm. Let's assume the contrary of the desired conclusion, namely for any open ball

U

there is an open ball

B_n \subset U

disjoint with

C_n

for any

n

Nested balls

So, there exist a closed ball

B(p_1, \epsilon_1/2)

with some radius

\epsilon > 0

and center

p_1

, which is disjoint with

C_1

, i.e.

B\left(p_1, \epsilon_1/2 \right) \cap C_1=\emptyset

.
Next, we can choose

p_2 \in B\left(p_1, \epsilon_1/ 4 \right)

, such that

B\left(p_2, \epsilon_2\right) \cap C_2=\emptyset

.
So, inductively there is a sequence

p_i, i=1, \ldots, k

and ball

B\left(p_k, \epsilon_1 / 2^k\right)

and

B\left(p_k, \epsilon_1 \ 2^k \right) \cap C_k=\emptyset

Convergent sequence

Thus, we have a sequence

\left\{p_k\right\}

X

. Since

d\left(p_{k+1}, p_k\right)<\epsilon_1 / 2^k

this is a Cauchy sequence, in fact

\begin{equation*}d\left(p_k, p_{k+l}\right)< \frac{\epsilon_1}{2^k}\end{equation*}

Since

X

is complete the Cauchy sequence converges to a some point

q \in X

, but by construction the point

q

not belonging to the union of

C_n

. So we get the condrattion

\begin{equation*}q \in X \quad \text{but} \quad q \notin \bigcup_{k=1}^n C_n\end{equation*}

This is the desired contradiction to the statement of theorem. Thus, at least one of the

C_n

must have an open ball of positive radius.

\Box

Second statement

The last assertion of the theorem is obvious from the first one applied to the closures of

A_n

Applications of Baire's category theorem

The uniform boundness principle

Theorem (The Uniform Boundedness Principle)
Suppose that

X

is a complete metric space and continuous functions

f_n : X \to \mathbb{R}

are such that for every

x \in X

, the sequence

\{f_n(x)\}

is bounded. Then there exists a closed ball

B

of some positive radius such that

\sup_n \sup_{x \in B} |f_n(x)| < \infty

\quad

Proof. Let

X_N := \{x \in X : \sup_n |f_n(x)| \leq N\}

. These sets are closed by the continuity of

f_n

. By hypothesis,

\begin{equation*}X = \bigcup_{N=1}^{\infty} X_N .\end{equation*}

According to Baire's theorem, some

X_N

has inner points and hence contains a closed ball

B

of a positive radius.

\Box

References

[Bogachev]
Bogachev, V. I., and O. G. Smolyanov. Real and Functional Analysis. Springer, 2020.

Baire category theorem

History and Motivation

Proof of Baire's category theorem

Applications of Baire's category theorem

History and Motivation

History and Motivation

Dense Sets on R\mathbb{R}R

Nowhere Dense Sets on R\mathbb{R}R

Proof of Baire's category theorem

Proof of Baire's category theorem

Nested balls

Convergent sequence

Second statement

Applications of Baire's category theorem

Applications of Baire's category theorem

The uniform boundness principle

References

References

Dense Sets on $\mathbb{R}$

Nowhere Dense Sets on $\mathbb{R}$