Geometry

1. Topology
   1. Basic Concepts
      1. Open and Closed Sets
         1. Definitions and examples
         2. Properties of open sets
         3. Properties of closed sets
         4. Interior, exterior, and boundary
         5. Limit points and closure of sets
      2. Continuous Functions
         1. Definition of continuity in terms of open sets
         2. Examples and counterexamples
         3. Continuous functions and homeomorphisms
         4. Intermediate Value Theorem in topological context
   2. Topological Spaces
      1. Definitions and Examples
         1. Definition of a topological space
         2. Bases and sub-bases
         3. Product topology
         4. Subspace topology
         5. Quotient topology
      2. Metric Spaces vs. Topological Spaces
         1. Definition of metric spaces
         2. Open balls and their role in metrics
         3. Comparison of metrics and general topology
      3. Convergence in Topology
         1. Convergence of sequences
         2. Net convergence
         3. Filters and ultrafilters
         4. Comparison with metric convergence
   3. Homeomorphisms and Topological Equivalence
      1. Definition of Homeomorphisms
         1. Properties of homeomorphisms
         2. Examples of homeomorphic spaces
      2. Topological Invariants
         1. Definition and importance
         2. Examples of invariants: connectedness, compactness, genus
      3. Simple vs. Complex Spaces
         1. Simple spaces: intervals, circles
         2. Complex spaces: tori, Möbius strips, Klein bottles
   4. Important Properties of Topological Spaces
      1. Compactness
         1. Definition and examples
         2. Heine-Borel Theorem
         3. Compactness in metric spaces
      2. Connectedness
         1. Definition and intuitive understanding
         2. Path-connectedness
         3. Connected components
         4. Applications in analysis and topology
      3. Separation Axioms
         1. T0, T1, T2 (Hausdorff) and higher separation axioms
         2. Relationship between separation properties
         3. Importance in defining space structure
   5. Specialized Topics in Topology
      1. Algebraic Topology
         1. Fundamental groups
         2. Homotopy and homology
         3. Applications in different fields
      2. Differential Topology
         1. Smooth manifolds
         2. Smooth maps and differentiable structures
         3. Tangent spaces
      3. Topological Groups
         1. Definition and examples
         2. Lie groups
         3. Applications in physics and algebra
      4. Knot Theory
         1. Classification of knots
         2. Invariants of knots
         3. Applications in DNA structure and molecular biology
      5. Point-Set Topology
         1. Detailed exploration of set-theoretic topology
         2. Zorn’s Lemma and applications
         3. Paracompactness and normal spaces
   6. Applications of Topology
      1. Science and Engineering
         1. Data analysis and topological data analysis
         2. Application in neuroscience for brain connectivity
      2. Computer Science
         1. Usage in network topology
         2. Algorithms for topology-based computations
      3. Medicine and Biology
         1. Topological methods in genomics and phylogenetics
         2. Analysis of biological networks and systems
      4. Economics and Social Sciences
         1. Topological models in economic theory
         2. Social network analysis
   7. Advanced Topics
      1. Topological Vector Spaces
         1. Function spaces and their topological properties
         2. Banach and Hilbert spaces from a topological view
      2. Functional Analysis
         1. Interaction with and contributions to topology
         2. Baire category theorem
         3. Applications in solving differential equations
      3. Homotopy Theory
         1. Basic definitions and concepts
         2. Homotopy groups and exact sequences
         3. Fibrations and cofibrations
      4. Morse Theory
         1. Critical points and topology of manifolds
         2. Applications in calculus of variations and dynamical systems