Baire Category Theorem
Key Definitions
Theorem (Baire Category Theorem)
Consequences
Nowhere Differentiable Functions
Category vs. Measure
Applications in Analysis
Proof that is not a countable union of lines.
Proof that there exist continuous functions that are monotone on no interval.
Proof of the open mapping theorem and closed graph theorem (in functional analysis).
Baire Spaces
Generic Properties
Generic properties are "typical" in the topological sense.
Example Applications
Linear Algebra:
- A generic matrix is diagonalizable
- A generic polynomial has distinct roots
Functional Analysis:
- A generic bounded linear operator is surjective
- A generic continuous function is nowhere monotone
Uniform Boundedness Principle
This is a consequence of the Baire category theorem.
Open Mapping Theorem
Closed Graph Theorem
Historical Note
The theorem was proved by RenΓ© Baire in 1899 and is fundamental to descriptive set theory and the study of "most" functions.