Lee in his book on topological manifolds says that
An $n$-manifold with a boundary is a second countable Hausdorff space in which any point has a neighborhood which is homeomorphic either to an open subset of $\mathbb R^n$ or to an open subset of $\mathbb H^n = \{x\in \mathbb R^n:x_n\geq 0\}$ endowed with a Euclidean topology.
As I understand, he means Euclidean topology which is a topology based on Eucledian metric and hence coincides with the subspace topology.
Isn't it equivalent to say that
An $n$-manifold with a boundary is a second countable Hausdorff space in which any point has a neighborhood which is homeomorphic to an open subset of $\mathbb H^n = \{x\in \mathbb R^n:x_n\geq 0\}$ endowed with a Euclidean topology.
As for me, since $\operatorname{Int}\mathbb H^n\approx \mathbb R^n$ by the homeomorphism $\varphi(x_1,...,x_n) = (x_1,...,\log x_n)$ these statements are equivalent, but maybe I'm missing something or there is another reason to put both $\mathbb H^n$ and $\mathbb R^n$ in the definition.
I guess my question does not itersects with About definitions of topological manifold with boundary since $\mathbb H^n$ is not homemorphic to $\overline{B_n}$ - closed unit ball in $\mathbb R^n$.