Neighbourhood in the definition of manifolds

Question

I hope my question is not too naive. Sorry in advance if it is.

When exploring the literature on topological manifolds, I found basically two main classes of definitions regarding the local "similarity" to $\mathbb{R}^n$. In the first definition, every point has an open neighbourhood which is homeomorphic to an open subset of $\mathbb{R}^n$. In other definitions, this neighbourhood is homeomorphic to $\mathbb{R}^n$ (or equivalently to the unit disk).

I am wondering on what extent these two characterizations are equivalent. It seems to me that they are not. They would be if any open subset of $\mathbb{R}^n$ was homeomorphic to $\mathbb{R}^n$, which is not the case (see for instance this question).

Is there something I'm missing here (for instance: is the equivalence induced by other properties of the manifold?) or are these two characterizations actually different?

Answer 1

Let $x\in M$ have an open neighbourhood $U$ homeomorphic to an open set $U'$ of $\Bbb R^n$. Then there is an open $V$ with $x\in V\subseteq U$ corresponding to an open disc in $U'$. Then $U$ is an open neighbourhood of $x$ homeomorphic to $\Bbb R^n$.