Topology

Online version:

General topology : Set theory and logic : Fundamental concepts ; Functions ; Relations ; The integers and the real numbers ; Cartesian products ; Finite sets ; Countable and uncountable sets ; The principle of recursive definition ; Infinite sets and the axiom of choice ; Well-ordered sets ; The maximum principle

Topological spaces and continuous functions : Topological spaces ; Basis for a topology ; The order topology ; The product topology on X x Y ; The subspace topology ; Closed sets and limit points ; Continuous functions ; The product topology ; The metric topology ; The metric topology (continued) ; The quotient topology

Connectedness and compactness : Connected spaces ; Connected subspaces of the real line ; Components and local connectedness ; Compact spaces ; Compact subspaces of the real line ; Limit point compactness ; Local compactness

Countability and separation axioms : The countability axioms ; The separation axioms ; Normal spaces ; The Urysohn lemma ; The Urysohn metrization theorem ; The Tietze extension theorem ; Imbeddings of manifolds

Tychonoff theorem : The Tychonoff theorem ; The stone-Cech compactification

Metrization theorems and paracompactness : Local finiteness ; The Nagata-Smirnov metrization theorem ; Paracompactness ; The Smirnov metrization theorem

Complete metric spaces and function spaces : Complete metric spaces ; A space-filling curve ; Compactness in metric spaces ; Pointwise and compact convergence ; Ascoli’s theorem

Baire spaces and dimension theory : Baire spaces ; A nowhere-differentiable function ; Introduction to dimension theory

Algebraic topology : The fundamental group : Homotopy of paths ; The fundamental group ; Covering spaces ; The fundamental group of the circle ; Retractions and fixed points ; The fundamental theorem of algebra ; The Borsuk-Ulam theorem ; Deformation retracts and homotopy type ; The fundamental group of () ; Fundamental groups of some surfaces

Separation theorems in the plane : The Jordan separation theorem ; Invariance of domain ; The Jordan curve theorem ; Imbedding graphs in the plane ; The winding number of a simple closed curve ; The Cauchy integral formula

The Seifert-van Kampen theorem : Direct sums of abelian groups ; Free products of groups ; Free groups ; The Seifert-van Kampen theorem ; The fundamental group of a wedge of circles ; Adjoining a two-cell ; The fundamental groups of the torus and the dunce cap

Classification of surfaces : Fundamental groups of surfaces ; Homology of surfaces ; Cutting and pasting ; The classification theorem ; Constructing compact surfaces

Classification of covering spaces : Equivalence of covering spaces ; The universal covering space ; Covering transformations ; Existence of covering spaces

Applications to group theory : Covering spaces of a graph ; The fundamental group of a graph ; Subgroups of free groups