Baire Category Theorem

The Baire Category Theorem states that complete metric spaces (and locally compact Hausdorff spaces) are "large" in the sense that they cannot be expressed as a countable union of nowhere-dense sets. This has profound consequences: continuous nowhere-differentiable functions exist, the rationals are "small" in the reals, and many existence proofs use Baire category.

Statement

Theorem8.1Baire Category Theorem

Let $(X, d)$ be a complete metric space. If $(U_n)$ is a countable collection of dense open subsets of $X$ , then $\bigcap_{n=1}^\infty U_n$ is dense in $X$ .

Equivalently: $X$ cannot be expressed as a countable union of nowhere-dense sets.

RemarkNowhere-dense

A set $A$ is nowhere-dense if its closure has empty interior: $\text{int}(\overline{A}) = \varnothing$ . For example, $\{0\}$ is nowhere-dense in $\mathbb{R}$ , as are $\mathbb{Z}$ and the Cantor set.

Applications

ExampleQ is meager in R

$\mathbb{Q} = \{q_1, q_2, q_3, \ldots\}$ is a countable union of singletons, each nowhere-dense. By Baire, $\mathbb{R}$ cannot equal $\mathbb{Q}$ (which we already knew). More subtly, $\mathbb{R} \setminus \mathbb{Q}$ is dense — "most" reals are irrational.

ExampleExistence of continuous nowhere-differentiable functions

The set of continuous functions on $[0, 1]$ that are differentiable at some point is a countable union of nowhere-dense sets (by Baire-based arguments). Thus by Baire, "most" continuous functions are nowhere differentiable — a counterintuitive but rigorous fact!

ExampleUniform Boundedness Principle

Baire is used to prove the Uniform Boundedness Principle in functional analysis: if $(T_n)$ is a sequence of bounded linear operators on a Banach space with $\sup_n \|T_n x\| < \infty$ for each $x$ , then $\sup_n \|T_n\| < \infty$ .

Summary

The Baire Category Theorem shows complete spaces are "large":

Cannot be written as countable union of nowhere-dense sets.
Countable intersections of dense open sets are dense.
Applications: existence of pathological functions, uniform boundedness, open mapping theorem.

See Completeness and functional analysis for more.