Noetherian and Irreducible Spaces
Explores hereditary and compactness properties of Noetherian spaces, irreducibility, finite irreducible decompositions, and examples inspired by prime numbers.
Topology
The Baire Category Theorem
Proves that a countable intersection of open dense subsets of a complete metric space is dense, using nested balls and a Cauchy-sequence construction.
The Topologist's Sine Curve
Analyzes the classical topologist's sine curve, proving connectedness through closure while showing that adjoining the origin does not produce a path-connected space.
Baire Category and Incomplete Normed Spaces
Uses the Baire category theorem to obtain local uniform boundedness for a pointwise bounded family of continuous functions, then proves a natural normed space of polynomials is incomplete.
Completion of Metric Spaces
Constructs the completion of a metric space from equivalence classes of Cauchy sequences, proves completeness and density of the canonical embedding, and examines completeness under homeomorphism.
Separation Axioms and Normal Spaces
Studies hereditary separation properties, Hausdorff quotient spaces formed by collapsing a closed set, and the theorem that regular second-countable spaces are normal.
The Tube Lemma and Closed Projections
Proves the tube lemma, closedness of projections with compact factors, a closed-graph criterion for continuity, and positive separation of disjoint closed sets when one is compact.
Number Theory
Arithmetic Topology on the Integers
Constructs a topology on the integers from arithmetic progressions, proves its basic sets are clopen, and derives the infinitude of primes topologically.
Euler's Totient Function
Develops multiplicativity and the prime-factor formula for Euler's totient function, then proves finiteness of its fibers and divisibility under divisors.
Gaussian Integers and Sums of Two Squares
Studies norms, units, conjugation, and prime splitting in the Gaussian integers, culminating in the criterion for representing an integer as a sum of two squares.
Complex Analysis
Injective Entire Functions
A proof that every injective entire function is affine, using Rouché's theorem, Cauchy estimates, singularities at infinity, and the open mapping theorem.