TITLE: Definition of a Manifold with a boundary
QUESTION [6 upvotes]: 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 $\mathbb R^n$.

REPLY [9 votes]: Yes they are equivalent. 
But his way of writing has pedagogical/expository advantages:

It becomes clear what is added compared to the definition of a manifold.
It paves the way to defining manifolds with corners as a further generalisation.
It allows one to make the statement that a boundary point is a point for which no neighborhood can be homeomorphic to an open subset of $\mathbb{R}^n$. 

Sometimes the most efficient way of writing things is not the most illuminating.