Definition of topological manifold, examples, coordinate chart and smoothly compatible coordinate charts.

A topological manifold $M$ of dimension $n$ is a topological space such that

1. $M$ is a Hausdorff space, for every pair of distinct points $x, y \in M$, there are disjoint open subsets $U, V \subset M$ such that $x \in U, y \in V$.
2. $M$ is second countable. There exists a countable basis for the topology of $M$. ¹⁾
3. $M$ is locally Euclidean of dimension $n$: each point of $M$ has a neighborhood that is homeormophic to an open subset of $\R^n$.

A line with two origins is an example of non-Hausdorff space but is locally Euclidean.

Let $M$ be a topological n-manifold. A coordinate chart on M is a pair $(U, \varphi)$, where $U$ is ane open subset of $M$ and $\varphi: U \to \hat U$ is a homeomorphism from $U$ to an open subset $\hat U = \varphi(U) \subset \R^n$. Let $p \in M$, if $U$ is a neighborhood of $p$ and $\varphi(p) = 0$, we say $(U, \varphi)$ is centered at $p$.

We follow the Example 1.4 in Lee for $S^n$.

To show that $S^n$ is an $n$-manifold, we only need to show it is locally Euclidean, since $S^n$ as a topological subspace of $\R^{n+1}$, it is automatically Hausdorff and second-countable.

In each of the $n+1$ axis directions in $\R^{n+1}$, we can cut the sphere $S^n$ in half, and get $$ U_i^+ = \{ \vec x \in S^n \mid x_i > 0 \} \quad U_i^- = \{ \vec x \in S^n \mid x_i < 0 \} $$ These gives $2n+2$ open sets on $S^n$. Clearly, they cover $S^n$.

We consider maps $$\gdef\B{\mathbb B} \pi_i^\pm: U_i^\pm \to \B^n, \quad (x_1, \cdots, x_{n+1}) \mapsto (x_1, \cdots, \hat x_i, \cdots, x_{n+1}) $$ where $\B^n \subset \R^n$ is the unit open ball in $\R^n$, and hat means omit that variable. We can check $\pi_i^+$ is a homeomorphism from $U_i^+$. Thus $(U_i^\pm, \pi_i^\pm)$ is a coordinate patch, for each $i=1, \cdots, n+1$.

Two charts $(U, \varphi)$ and $(V, \psi)$ are said to be smoothly compatible if either $U \cap V = \emptyset$, or $\psi \circ \varphi^{-1}: \varphi(U\cap V) \to \psi(U \cap V)$ is a diffeomorphism.

? Let $f: A \to B$ be a homeomorphism of open subsets in $\R^n$. If $f$ is smooth, is it automatically true that $f^{-1}$ is smooth?

$$\gdef\acal{\mathcal A}$$ An atlas for $M$ is a collection of charts whose domains cover $M$. An atlas $\acal$ is smooth, if any two charts in it are smoothly compatible with each other.

See this note about difference between topological structure and differential structure. https://www3.nd.edu/~lnicolae/FYsem2003.pdf