Topological manifolds are defined to be locally Euclidean (e.g. John Lee). That is, any point is in an open set that is homeomorphic to either $\mathbb{R}^n$, an open ball in $\mathbb{R}^n$ or an open subset of $\mathbb{R}^n$.
I understand why "$\mathbb{R}^n$" and "open ball in $\mathbb{R}^n$" are equivalent definitions of locally Euclidean: any open ball is homeomorphic to $\mathbb{R}^n$, and the composition is again a homeomorphism. But why is "open subset of $\mathbb{R}^n$" also equivalent? Since an open subset that is not connected is not homeomorphic to any open balls.
Attempt: suppose $p$ is a point on the manifold and $U$ is the open set containing $p$ that is homeomorphic to an open subset $V$ of $\mathbb{R}^n$. There is an open ball centered at the homeomorphic image of $p$ in $V$.
The inverse image of this open ball is an open set $W$ in the manifold. Therefore, $p$ is in an open set $W$ that is homeomorphic to an open ball (by restricting the previous homeomorphism to the inverse image). For the other direction: an open ball is an open set.
