Basis to a manifold by coordinate balls I was trying to show that "Every manifold has a basis of coordinate balls". My approach was like this: Given a manifold $M$, it's well-known that for every point $\mathbf{x}\in M$ there exists a neighborhood of $\mathbf{x}$ homeomorphic to an open ball in $\mathbb{R}^n$. At first, I thought that I could obtain this basis using the set of all balls in $\mathbb{R}^n$ that are homeomorphic to some neighborhood of every point in $M$, but this became confusing to me. Is this the right approach? If yes, how can I go on?
 A: Hints:
Question 1:
Yes, that's probably the right approach.  Here's a generous hint to help you get unstuck: Use the definition of a basis for a topology, in conjunction with some fact about restriction of coordinate charts.
Question 2:
As for your second question, what are the standard coordinate charts on $\mathbb{S}^n$?  To be concrete, work with $n=2$.  What is the closure of such a chart?

Answers follow.
Question 1:
Let $U$ be an open set in $M$.  Let $x\in U$.  About $x$ there exists some open neighborhood $C$ which admits a diffeomorphism $\phi_C$ to some open neighborhood of the origin of $\mathbb{R}^n$.  Let $B$ denote the component of $U\cap C$ which contains $x$.  Then $\phi_C|_B$ is a diffeomorphism onto its image, which is again a connected open neighborhood of the origin of $\mathbb{R}^n$.  (By possibly postcomposing with a translation, we may WLOG assume that $\phi_C(x) = 0$.)  Now take a small open ball $S$ about $0$ in $\mathbb{R}^n$ which has $0\in S\subseteq \phi_C(B)$.  Its preimage $\phi_C^{-1}(S)$ is diffeomorphic to $\mathbb{R}^n$ via $\phi_C|_{\phi_C^{-1}S}\circ \varphi$, where $\varphi$ is a diffeomorphism between an $\epsilon$-ball and all of $\mathbb{R}^n$, so $\phi_C^{-1}(S)$ together with the restriction of $\phi_C$ is a coordinate chart.  Finally we have $x\in\phi_C^{-1}(S)\subseteq U$.
Therefore, for every open set $U$ of $M$ and for every point $x\in U$ we may find a coordinate chart $S$ which has $x\in S\subseteq U$, so coordinate neighborhoods form a basis for the topology of $M$.
Question 2:
Use stereographic projection to get coordinates.  Equivalently, $\mathbb{S}^n$ is the one-point compactification of $\mathbb{R}^n$.
