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I need some clarifications on topological manifolds. My professor has defined them as second countable, connected, Hausdorff topological spaces which are locally homeomorphic to $\mathbb{R}^n_+$, where $\mathbb{R}^n_+=\{(x_1,\dots,x_n)\in \mathbb{R^n}|x_n>0\}$.

My question is: why do we ask that the target of the local homeomorphisms is $\mathbb{R}^n_+$ and not simply $\mathbb{R}^n?$

batman
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    @Clarreta With your definition both would be exactly the same, but one generally defines it (when dealing with manifolds with boundary) to be locally homeomorphic to the upper half-space ${(x_1, \dots, x_n) \in \mathbb{R}^n:, x_n \geq 0}$, and in this case it is different. – i like xkcd Jan 09 '14 at 16:31
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    Also, it's highly unusual to include connectedness in the definition of manifolds. The statements of a lot of theorems are simpler if manifolds are allowed to be disconnected. – Jack Lee Jan 09 '14 at 21:56

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