Every point in a k-manifold has a neighborhood diffeomorphic to $\Bbb{R}^k$

Question

The problem comes from Alan Pollack's Differential Topology, pg. 5. Suppose that X is a k-dimensional manifold. Show that every point in X has a neighborhood diffeomorphic to all of $\Bbb{R}^k$.

I have already shown that $\Bbb{R}^k$ is diffeomorphic to $B_a$ (part (a) of the question) the open ball of radius $a$, though have little to no understanding of how to proceed.

Thank you

The author defines a k-manifold as a set such that each point possesses a neighborhood diffeomorphic to an open set of $\Bbb{R}^k$

Answer 1

You already did all the work. If $x\in X$, then there is a neighborhood $U$ of $x$ that is diffeomorphic to an open set of $\mathbb{R}^k$. Take $\phi: U\to V$ to be the chart. Since $V$ is open, there is an open ball $B_a$ centered at $x$ contained in $V$. Take $W=\phi^{-1}(B_a)$. Then $W$ is diffeomorphic to $B_a$, and as you already proved, this is diffeomorphic to $\mathbb{R}^k$.

Answer 2

Given a point $p \in \mathbb{R}^k$, and an open neighbourhood $U$ of $p$, there exists an open set $V$ with $p \in V \subset U$ such that $V$ is diffeomorphic to the $k$-dimensional unit ball. Just take the Euclidean metric on $\mathbb{R}^k$, and let $V$ be a sufficiently small open ball centred on $p$.