# Wednesday, August 17

:::{.remark}
Main objects of study: smooth manifolds, a class of topological spaces on which Calculus can naturally be carried out.
Some examples of things that *should* be smooth manifolds:

- $\RR^n$ as we've learned in multivariable Calculus,
- $S^2$, since we generally believe Calculus works on the planet Earth.[^earth_sphere]

A precise definition will be that a smooth manifolds is a *topological manifold* with the extra structure of a *smooth atlas*.

[^earth_sphere]: 
Which is approximately a sphere.

:::

:::{.definition title="Topological manifolds"}
A **topological manifold** of dimension $n$ is a topological space $M$ such that

- $M$ is Hausdorff (points are separated by opens),
- $M$ is second-countable (admitting a countable base),
- $M$ is locally homeomorphic to $\RR^n$, or equivalently some open subset thereof (i.e. $M$ admits continuous charts).

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:::{.exercise title="?"}
Show that $\RR^n$ and $S^2$ are topological manifolds, and construct charts for $S^2$.
Note that spherical coordinates 
\[
(0, \pi)\times (0, 2\pi) &\to S^2 \\
(\phi, \theta) & \mapsto \tv{\sin\phi\cos\theta, \sin\phi\sin\theta, \cos\theta}
x\]
misses only a meridian and the poles:

\begin{tikzpicture}
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\end{tikzpicture}

An alternative is stereographic projection:
\[
S^2\smts{\tv{0,0,1}} &\to \RR^2 \\
\tv{x,y,z} &\mapsto \tv{{x\over 1-z}. {y\over 1-z}}
.\]

\begin{tikzpicture}
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\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2022/Fall/DiffGeo/sections/figures}{2022-08-17_12-13.pdf_tex} };
\end{tikzpicture}


> Note that a random subset of $\RR^n$ will likely not homeomorphic to any $\RR^d$!

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:::{.example title="Non-Hausdorff 'manifolds'"}
Take the line with two origins $(\RR\times\ts{0}) \disjoint (\RR\times\ts{1})/ (x,0)\sim (x, 1)$ for all $x\neq 0$.
Any neighborhood about the origin is open since its preimage is open:

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![](figures/2022-08-17_12-21-38.png)

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