The Baire category theorem is a foundational theorem of analysis. It states that a complete metric space cannot be a countable union of nowhere dense sets. This theorem converts completeness into a strong largeness principle and supports many applications in real analysis and functional analysis. :::example[Van der Waerden Function] Let $h$ be the triangular periodic function on $\mathbb{R}$ and set $f(x)=\sum_{n=0}^{\infty}h(10^n x)/10^n$. This series defines a continuous nowhere differentiable function. ::: :::theorem[Baire Category Theorem] Every complete metric space is of category II. Equivalently, a complete metric space cannot be written as a countable union of nowhere dense subsets. ::: :::proof Given that $(X,d)$ is complete. To prove that $X$ is of category II. Suppose $X=\bigcup_{n=1}^{\infty}E_n$ with each $E_n$ nowhere dense. Construct nested nonempty closed balls $B_n$ such that $B_{n+1}\subset B_n$, $B_n\cap E_n=\varnothing$, and $\operatorname{diam}(B_n)<1/n$. This is possible because each $E_n$ is nowhere dense. By the Cantor intersection theorem, $\bigcap_{n=1}^{\infty}B_n$ contains a point $x$. But $x\notin E_n$ for every $n$, contradicting $X=\bigcup_{n=1}^{\infty}E_n$. Hence $X$ is not of category I. $\square$ ::: :::solved-problem State and prove the local boundedness conclusion for a pointwise bounded family of continuous functions on $\mathbb{R}$. ::: :::solution For each $n$, let $E_n=\{x:|f(x)|\leq n\text{ for all }f\in\mathcal{F}\}$. Each $E_n$ is closed and $\mathbb{R}=\bigcup E_n$. By Baire, some $E_N$ has nonempty interior. On $Y=E_N^\circ$, every $f\in\mathcal{F}$ satisfies $|f(x)|\leq N$. $\square$ ::: :::exercise [1] State the main definition from this lesson. [2] State the main theorem from this lesson. [3] Give one example illustrating the theorem. [4] Explain one common mistake related to this topic. ::: :::answer [1] The main definition is the first titled definition in the lesson. [2] The main theorem is the titled theorem proved in the lesson. [3] The examples in the lesson provide standard choices. [4] A common mistake is to ignore the metric or the ambient space when applying the definition. ::: :::faq[Frequently Asked Questions] Q: Why is the ambient metric space important? A: Openness, closedness, closure, and boundary are interpreted relative to the chosen metric space. Q: Can the same set behave differently in different metric spaces? A: Yes. Changing the metric or the ambient space may change topological behaviour. Q: How should students check a definition? A: They should return to balls, closures, and sequences, depending on the context. ::: :::call-to-action[Continue from Baire Category Theorem and Applications] subtitle: Review the definitions and proofs before moving through the metric topology sequence. button: Next Lesson | /functional-analysis/metric-spaces/topological-structure-of-metric-spaces/oscillation-and-sets-of-discontinuity button-ghost: Previous Lesson | /functional-analysis/metric-spaces/topological-structure-of-metric-spaces/category-and-nowhere-dense-sets :::