Faculty evidence

Lesson Keys

Use keys to evaluate reasoning, not merely final statements. Accept equivalent valid proofs.

U01 L01

Why Topology? Space Beyond Distance

U01 L02

Topological Spaces and the Open-Set Axioms

U01 L03

Closed Sets, Complements and Dual Axioms

U01 L04

Neighborhoods, Interior, Closure and Boundary

U01 L05

Bases and Local Descriptions of Topology

U01 L06

Subbases and Generated Topologies

U01 L07

Continuity Through Preimages

U01 L08

Homeomorphisms and Topological Equivalence

U02 L01

Metrics, Open Balls and Induced Topologies

U02 L02

Convergence, Cauchy Sequences and Completeness

U02 L03

Equivalent Metrics and Topological Invariance

U02 L04

First Countability and Sequence Tests

U02 L05

Second Countability, Separability and Lindelöf Ideas

U02 L06

Separation Axioms from T₀ to Hausdorff

U02 L07

Regularity, Normality and Metrization Foundations

U03 L01

Compactness by Open Covers

U03 L02

Compactness in Metric and Euclidean Spaces

U03 L03

Continuous Images and Extreme-Value Consequences

U03 L04

Connectedness and Separation

U03 L05

Path Connectedness and Components

U03 L06

Product Spaces and Universal Coordinates

U03 L07

Quotient Spaces, Identifications and Gluing

U04 L01

Homotopy as Continuous Deformation

U04 L02

Loops, Concatenation and Path Homotopy

U04 L03

The Fundamental Group

U04 L04

Simply Connected Spaces and Contractibility

U04 L05

Seifert–van Kampen Strategy

U04 L06

Covering Spaces and Lifting

U04 L07

Deck Transformations and Classification Examples

U05 L01

Simplicial Complexes and Triangulated Spaces

U05 L02

CW Complexes and Cell Attachments

U05 L03

Chains, Boundaries and the Identity ∂²=0

U05 L04

Homology Groups and Dimensional Holes

U05 L05

Exact Sequences and Mayer–Vietoris Strategy

U05 L06

Euler Characteristic and Homological Counting

U05 L07

Cohomology, Products and Duality Preview

U06 L01

Topological Manifolds and Local Euclidean Structure

U06 L02

Charts, Atlases and Smooth Compatibility

U06 L03

Smooth Maps and Diffeomorphisms

U06 L04

Tangent Vectors as Velocities and Derivations

U06 L05

Differentials, Rank and the Chain Rule

U06 L06

Cotangent Spaces, Covectors and Tensors

U06 L07

Vector Fields, Flows and Lie Brackets

U07 L01

Inverse and Implicit Function Theorems on Manifolds

U07 L02

Immersions, Submersions and Embeddings

U07 L03

Regular Values and the Preimage Theorem

U07 L04

Critical Points, Sard’s Theorem and Genericity

U07 L05

Transversality and Stable Intersections

U07 L06

Orientation, Degree and Winding

U07 L07

Vector-Field Index and Poincaré–Hopf Preview

U08 L01

Regular Curves and Arc-Length Parametrization

U08 L02

Curvature, Torsion and the Frenet Frame

U08 L03

Local and Global Geometry of Plane Curves

U08 L04

Parametrized Surfaces and Tangent Planes

U08 L05

The First Fundamental Form

U08 L06

The Second Fundamental Form and Shape Operator

U08 L07

Gaussian Curvature, Mean Curvature and Theorema Egregium

U09 L01

Riemannian Metrics and Smooth Inner Products

U09 L02

Length, Energy and Riemannian Distance

U09 L03

Connections and Covariant Differentiation

U09 L04

The Levi–Civita Connection

U09 L05

Geodesics and the Geodesic Equation

U09 L06

Exponential Maps, Normal Coordinates and Completeness

U09 L07

Riemann, Sectional, Ricci and Scalar Curvature

U10 L01

Alternating Tensors and the Exterior Algebra

U10 L02

Differential Forms and Pullbacks

U10 L03

The Exterior Derivative

U10 L04

Integration on Oriented Manifolds

U10 L05

Generalized Stokes Theorem

U10 L06

Closed, Exact and de Rham Cohomology

U10 L07

Poincaré Lemma and de Rham Correspondence

U11 L01

Vector Bundles, Local Triviality and Sections

U11 L02

Transition Functions and Bundle Gluing

U11 L03

Tangent, Cotangent and Tensor Bundles

U11 L04

Lie Groups, Lie Algebras and Exponential Maps

U11 L05

Principal Bundles and Gauge Symmetry

U11 L06

Connections, Curvature Forms and Holonomy

U11 L07

Characteristic-Class Foundations

U12 L01

Morse Functions and Nondegenerate Critical Points

U12 L02

Handles, Cell Attachments and Morse Inequalities

U12 L03

Gauss–Bonnet for Surfaces

U12 L04

Poincaré–Hopf and Vector-Field Indices

U12 L05

Jacobi Fields and Geodesic Variation

U12 L06

Comparison Geometry and Curvature Bounds

U12 L07

Global Rigidity, Finiteness and Research Frontiers

U13 L01

Classification of Compact Surfaces

U13 L02

Knots, Links and Reidemeister Moves

U13 L03

Computational Topology and Persistent Homology

U13 L04

Geometric Flows and Shape Evolution

U13 L05

Geometry in Relativity, Gauge Theory and Physics

U13 L06

Research Design, Literature and Reproducibility

U13 L07

Capstone Synthesis and Oral Defense