Possible Duplicate:
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.