Foundations of advanced analysis · Graduate course
General Topology
The structural language of continuity, convergence, and space.
Topology is the study of what survives continuous deformation: nearness without distance, continuity without formulas, and space stripped to its structural essentials. Starting from metric intuition, this course builds the axioms of a topological space and develops bases, closure and interior, continuity and homeomorphism, the subspace, product, and quotient constructions, and the great structural theorems of connectedness, compactness, and separation, closing with nets, filters, and the topological foundations that analysis rests on.
Where this course sits
What you will master
Three capabilities the course builds
Reason about continuity, convergence, and nearness through open sets alone, independent of any distance or formula.
Use connectedness, compactness, and separation to prove global results, from the intermediate value theorem to Tychonoff and Urysohn.
Establish the topological foundations that measure theory, distribution theory, complex analysis, and functional analysis all depend on.
The syllabus
Fourteen chapters, built in sequence
Each chapter opens with visual investigations, states every definition and theorem with complete proofs, and closes with a graded set of exercises carrying progressive hints and full corrections.
Functions, injections, surjections, and bijections, the double-inclusion proof technique, inverse images, and the algebraic foundation on which topology is built.
Open chapterDistance functions, open and closed balls, and the metric topology. Convergence and completeness in metric spaces as a bridge to the abstract definition.
Open chapterThe axioms of a topology, open and closed sets, and the three canonical topologies. The discrete, indiscrete, and cofinite topologies as reference points.
Open chapterGenerating topologies from a basis or sub-basis, the topology generated by a collection, and comparing topologies by fineness.
Open chapterThe derived set operators and their interplay, dense and nowhere-dense subsets, and the Cantor set as an extreme example.
Open chapterThe open-set definition of continuity, homeomorphisms as topological isomorphisms, topological invariants, and the classification problem.
Open chapterThe induced topology on a subset, relative openness and closedness, and open and closed embeddings and their role in analysis.
Open chapterThe Tychonoff product topology and its universal property, projection maps, products of Hausdorff spaces, and the diagonal and separation.
Open chapterIdentification spaces and gluing, quotient maps and the universal property, and building the circle, torus, Klein bottle, and projective plane from squares.
Open chapterClopen sets, connected components, and path-connectedness, the intermediate value theorem from topology, and the topologist's sine curve.
Open chapterOpen covers and finite subcovers, Heine-Borel, the Extreme Value theorem and the Tube Lemma, and why compact Hausdorff spaces are normal.
Open chapterThe separation hierarchy from T0 to T4, Urysohn's Lemma and Tietze Extension, first and second countability, Lindelof spaces, and Urysohn metrisation.
Open chapterWhy sequences are insufficient in general spaces, nets and filters as the correct convergence language, and ultrafilters with Tychonoff's theorem.
Open chapterThe Baire Category Theorem, one-point and Stone-Cech compactifications, the Riesz representation theorem, and the generic existence of nowhere differentiable functions.
Open chapterFour premium investigations per chapter, on light crisp backgrounds.
Every theorem stated with all hypotheses and proved in full.
Exercises with progressive hints and a full worked correction.
Full English and French presentation sharing one mathematical corpus.