Topology

1. Advanced Topics
   1. Topos Theory
      1. Category Theory Connections
         1. Basic Definitions in Category Theory
            1. Categories, Functors, and Natural Transformations
            2. Limits and Colimits
            3. Adjunctions
         2. Topos as a Generalized Space
            1. Topos Theory Fundamentals
               1. Definition of a Topos
               2. Examples: Set, PreSheaf, Sheaf Topos
            2. Geometric Morphisms
               1. Direct and Inverse Image Functors
               2. Monadicity of Direct Image Functors
      2. Grothendieck Topology
         1. Definition and Examples
            1. Sites and Sheaves
            2. Sheafification Process
         2. Applications in Algebraic Geometry
            1. Étale Topology
            2. Cohomological Techniques
         3. Covering Sieves and Descent
            1. Types of Covering Families
            2. Effective Descent
   2. Differential Topology
      1. Smooth Maps
         1. Differentiable Maps and Functions
            1. Definitions: Smooth Manifolds and Maps
            2. The Smooth Structure
         2. Sard's Theorem
            1. Statement and Proof Outline
            2. Applications and Consequences
         3. Transversality
            1. Intersection Theory
            2. Transversality Theorem
      2. Morse Theory
         1. Critical Points and Morse Functions
            1. Definition and Examples
            2. Regular Values Theorem
         2. Morse Inequalities
            1. Statement and Proof
            2. Consequences for Topological Classification
         3. Handle Decomposition
            1. Attaching Maps and Handlebodies
            2. Cobordism
   3. Set-Theoretic Topology
      1. Cardinal Functions
         1. Cardinality Concepts in Topology
            1. Weight, Density, and Character
         2. Impact on Space Properties
            1. Relationship to Separation Axioms
      2. Lindelöf Spaces
         1. Definitions and Examples
            1. Countability Properties
            2. Relations to Other Compactness Concepts
         2. Lindelöfness in Product Spaces
      3. Baire Category Theorem
         1. Baire Spaces and Applications
            1. Complete Metric Spaces
            2. Implications for Function Spaces
         2. Historical Context and Proof
   4. Point-Set Topology
      1. Basic Definitions and Results
         1. Topological Properties
            1. Open and Closed Sets
            2. Bases and Subbases
         2. Topological Constructs
            1. Product and Quotient Topologies
            2. Initial and Final Topologies
      2. Continuity and Convergence
         1. Reminders of Continuous Functions
            1. Cone Criterion
            2. Subspace Continuity
         2. Convergence of Sequences, Nets, and Filters
            1. Definitions and Relations
            2. Limits and Cluster Points