Suppose a manifold $M$ is connected. (Here I assume that a manifold is a Hausdorff, second countable space and each point $x\in M$ has a neighbourhood homeomorphic to $\mathbb{R}^{n}$, where $n$ can depend on $x$ in general)
How do I show that there exist $n$ such that each point in $M$ has a neighborhood homeomorphic to $\mathbb{R}^{n}$?
Let $A_n$ be the set of all points $x\in M$ such that $x$ has a neighborhood homeomorphic to $\mathbb R^n$. It is easy to see that each $A_n$ is open. Choose $n$ such that $A_n$ is nonempty. If $A_n^C$ is nonempty, then show why $A_n$ and $A_n^C$ form a separation. This is a contradiction, therefore $A_n=M$.