## 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 $X$.