Topology

Topology is a major area of mathematics concerned with the properties of geometric objects that are preserved under continuous deformations, such as stretching, bending, and twisting, but not tearing or gluing. Often called "rubber sheet geometry," this field ignores rigid metric properties like distance and angles to focus instead on qualitative attributes like connectivity, compactness, and, most famously, the number of holes in an object. For this reason, a coffee mug is topologically equivalent to a donut (a torus), as both possess a single hole, an invariant property that allows one to be smoothly reshaped into the other.

  1. Introduction to Topological Concepts
    1. Intuitive Understanding of Topology
      1. The Rubber Sheet Geometry Analogy
        1. Invariance under Continuous Deformation
          1. Distinction from Euclidean Geometry
            1. Distinction from Analysis
              1. Topological Equivalence
                1. Examples of Topologically Equivalent Objects
                  1. Donut and Coffee Mug
                    1. Sphere and Cube
                      1. Disk and Square
                    2. Historical Development
                      1. Early Topological Problems
                        1. Euler's Solution to the Königsberg Bridge Problem
                          1. Contributions of Henri Poincaré
                            1. Development of Point-Set Topology
                              1. Emergence of Algebraic Topology
                                1. Modern Applications and Connections
                                2. Foundational Set Theory
                                  1. Basic Set Concepts
                                    1. Sets and Elements
                                      1. Set Notation and Conventions
                                        1. Subsets and Proper Subsets
                                          1. Power Sets
                                            1. Partitions of Sets
                                            2. Set Operations
                                              1. Union of Sets
                                                1. Intersection of Sets
                                                  1. Set Difference
                                                    1. Complement of Sets
                                                      1. Symmetric Difference
                                                        1. Cartesian Product
                                                          1. Properties of Set Operations
                                                          2. Indexed Families of Sets
                                                            1. Definition and Notation
                                                              1. Arbitrary Unions and Intersections
                                                                1. De Morgan's Laws for Indexed Families
                                                                2. Relations
                                                                  1. Definition of Relations
                                                                    1. Properties of Relations
                                                                      1. Reflexivity
                                                                        1. Symmetry
                                                                          1. Transitivity
                                                                            1. Antisymmetry
                                                                            2. Equivalence Relations
                                                                              1. Equivalence Classes
                                                                                1. Partial Orders
                                                                                  1. Total Orders
                                                                                  2. Functions
                                                                                    1. Definition of Functions
                                                                                      1. Domain and Codomain
                                                                                        1. Range and Image
                                                                                          1. Injective Functions
                                                                                            1. Surjective Functions
                                                                                              1. Bijective Functions
                                                                                                1. Inverse Functions
                                                                                                  1. Composition of Functions
                                                                                                  2. Cardinality
                                                                                                    1. Finite Sets
                                                                                                      1. Infinite Sets
                                                                                                        1. Countable Sets
                                                                                                          1. Uncountable Sets
                                                                                                            1. Cantor's Diagonal Argument
                                                                                                              1. Comparison of Cardinalities