differential topology definitions

This is just a bunch of definitions from Guillemin and Pollack

Smooth A mapping $f$ of an open set $U \subset \mathbb{R}^n$ into $\mathbb{R}^m$ is called smooth if it has continuous partial derivatives of all orders.

Local In the neighborhood of a point.

Diffeomorphism A smooth mapping $f$ is a diffeomorphism if it is bijective and the inverse $f^{-1}$ is smooth.

Manifold A subset $X \subset \mathbb{R}^n$ is a k-dimensional manifold if it is locally diffeomorphic to $\mathbb{R}^k$

Parametrization A parametrization of a manifold is a diffeomorphism $\phi : U \to \mathbb{R}^n$ where $U$ is an open neighborhood in manifold.

Coordinate System A coordinate system is the inverse diffeomorphism of a parametrization.

Submanifold If $Z$ and $X$ are both manifolds in $\mathbb{R}^n$ and $Z \subset X$, then $Z$ is a submanifold of $X$. Any open subset of $X$ is a submanifold of $X$.