General (Point-Set) Topology
- Topological Spaces - Open sets, bases, subbases
- Neighborhoods and Limit Points
- Continuity and Homeomorphisms
- Separation Axioms
- T0, T1, T2 (Hausdorff)
- Regular and Normal Spaces
- Countability Axioms
- First and Second Countable Spaces
- Separability
- Compactness
- Heine-Borel Theorem
- Sequential Compactness
- Local and σ-Compactness
- Compactification (Alexandroff, Stone–Čech)
- Connectedness
- Path-Connectedness
- Connected Components
- Product, Subspace, and Quotient Topologies
- Baire Category Theorem
- Urysohn’s Lemma & Tietze Extension Theorem
Metric & Normed Spaces
- Metric Spaces
- Inner Product Space
- Normed Vector Spaces
- Banach Spaces
- Hilbert Spaces
- Completions of Metric Spaces
- Contraction Mapping / Banach Fixed-Point Theorem
Algebraic Topology
- Homotopy Theory
- Homotopy Equivalence
- Fundamental Group
- Higher Homotopy Groups
- Covering Spaces
- Homology Theory
- Simplicial Homology
- Singular Homology
- Cellular Homology
- Cohomology
- CW Complexes
- Euler Characteristic
Differential Topology
- Manifolds
- Smooth Manifolds, Charts, Atlases
- Manifolds with Boundary
- Tangent Spaces and Tangent Bundles
- Differential Forms
- Vector Fields and Flows
- Sard’s Theorem, Degree Theory
Functional-Analytic Topology
- Weak and Weak-* Topologies
- Compact Operators
- Dirichlet Forms
- Infinite-Dimensional Topological Vector Spaces
Applications
- Topological Data Analysis - Persistent Homology
- Stochastic Topology and Random Geometry - Random graphs, geometric complexes
- Topology in Dynamical Systems
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