Baire Category Theorem
Complete metric spaces are not countable unions of nowhere dense sets.
Key idea
Complete metric spaces are not countable unions of nowhere dense sets.
In depth
In a complete metric space, a countable union of nowhere dense sets cannot fill the space. Consequently, a complete space is not a countable union of "thin" sets. It proves the existence of continuous nowhere differentiable functions and underlies major functional analysis results.
Worked example
R cannot be written as a countable union of nowhere dense sets, so it is uncountable.
Real-world uses
- A tool for proving the existence of functions with unusual properties in analysis
Common mistake
Applying it to incomplete spaces like Q.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Baire Category Theorem?
Complete metric spaces are not countable unions of nowhere dense sets. It is a topic at the expert level in Analysis & Topology.
Where is Baire Category Theorem used in real life?
Real-world examples: A tool for proving the existence of functions with unusual properties in analysis.
What should I learn before Baire Category Theorem?
Make sure you are comfortable with Banach Fixed Point Theorem, the previous topic in Analysis & Topology.
What is a common mistake with Baire Category Theorem?
Applying it to incomplete spaces like Q.
How can I practise Baire Category Theorem?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Baire Category Theorem?
Move on to Arzelà–Ascoli Theorem next.
Related expert topics in Analysis & Topology
- Axioms of the Real Numbers – ℝ is a complete ordered field.
- Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
- Archimedean Property – For any real x there is an integer n > x.
- Density of Rationals – Between any two reals there is a rational.
- Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
- Limit Theorems for Sequences – Algebra of limits and squeeze theorem.