# Definition of a Manifold with a boundary

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 $\mathbb R^n$.

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No, I don't think there's a difference. He probably just wants all of his old examples of manifolds to hold up without (minor, granted) change. – Dylan Moreland Oct 19 '11 at 14:25
@DylanMoreland Thanks, that may be the case. I don't think that question needs more extended answer, so I will be happy to admit your comment if you put it as an answer. Oh, I'm sorry, Willie was the first. – Ilya Oct 19 '11 at 14:42
@joriki: thank you for flagging the duplicate - I just noticed that there are two identical question on the list on my profile page. – Ilya Oct 19 '11 at 15:39

Yes they are equivalent.

But his way of writing has pedagogical/expository advantages:

1. It becomes clear what is added compared to the definition of a manifold.
2. It paves the way to defining manifolds with corners as a further generalisation.
3. 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.

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Thank you very much, Willie – Ilya Oct 19 '11 at 14:44
In fact, I'm reading the book from 2011 while in the 2001 version of that book Lee introduced it only through $\mathbb H^n$. – Ilya Oct 19 '11 at 16:18