Manifolds Lecture Notes (Exam 1)

Metric and metric space Definitions

Ball Definition

Open and closed in a metric space Definitions

Metric Topology Definition and Restriction “Prop”

What is open in a metric subspace? Prop

Continuous function in a metric space Definition

function is continuous iff Prop

continuous iff (closed sets) Prop

homeomorphism definition + relationship to open sets prop

Open cover and compactness Definitions

bounded Definition and Heine Borel Theorem

If a subset of an arbitrary metric space is compact.. Prop

If a subset of the domain of a continuous function is compact… Theorem

If you have a continuous function and a compact subset of the domain… **Corollary

if a you have a function that’s continuous and injective, and that maps a compact metric space to an arbitrary metric space… Prop

a closed subset of a compact metric space is... Lemma

Path Definition

Path-Connected Definition

path from p to q relation… Claim

Path components Definition

path connected and connected are equivalent for manifolds.

If a subset of the domain of a continuous function is path… Prop

Smooth on open subsets of Rn Definition

smooth on sets of Rn

Diffeomorphism from subset of Rn to subset of Rm Definition

k dimensional manifold, local parametrization, and coordinate chart Definitions

Differentiable Definition

If f is differentiable at p, then for all v in Rn… Theorem

Uniqueness of Derivatives Theorem

Differentiability and Continuity Theorem

Differentiability, Partial Derivatives and Jacobian Matrix

Suppose every first order partial derivative of f exists in an open neighborhood at p…. Theorem

Chain Rule Theorem