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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.

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    It seems you accidentally posted this twice? I'm marking this one as the duplicate, even though technically it's the earlier one, because you added two sentences to the other one and it's already been upvoted once.2011-10-19

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