Manifolds, charts and atlases

Transition maps

Mapping 2D manifolds to Riemann surfaces

Refinement

Ply (order) of a cover

Small inductive dimension

Large inductive dimension

Lebesgue covering dimension

Paths and loops

Holes and genuses

Path-connect spaces

Elliptic (Riemann sphere)

Parabolic (complex plane)

Hyperbolic (open disk)

Torus

Hyper-elliptic curves

Functions between topologies

Homotopy

Homeomorphisms

Vector bundles

Bundle projection

Trivial and twisted bundles

Cross-sections and zero-sections of fibre bundles

Trivial bundles and the torus

Twisted bundles and the Klein bottle

Mobius strips

Submanifolds

Boundries and interiors

Closed manifolds

Mapping 2D manifolds to Riemann surfaces (needs to be orientable and metricisable)