#diffgeo #topology
## Definition
An $n$-dimensional *smooth manifold* $M$ is a [[Topology|topological space]] that is
- paracompact (it has a locally finite cover)
- Hausdorff (point-separable)
- locally euclidean: For all $p \in M$ there is some neighborhood $U_p \ni p$ and a Map $f_p: U_p \to V_p$ to some open subset $V_p \subseteq \field{R}^n$ that is [[Smooth Map|smooth]].
If $f_p$ is only $r$-times differentiable, we call $M$ a $C^r$-manifold. If $f_p$ is only continuous, we call $M$ a *topological manifold*.
## Properties
From paracompact it follows: An open subset of a manifold has at most countably many connected components.
Locally euclidean is equivalent to the existence of a smooth [[Atlas]].
[[Lie Group]]s are manifolds.
## Examples
- $\mathbb{R}^n$ itself
- Product spaces of manifolds
- $\mathbb{S}^1$
- $n$-Tori