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Let $X$ be a topological Hausdorff space with countable basis, let $m \in \mathbf{N}$. Let us consider two definition of topological manifold with boundary.

Definition 1. $X$ is called a $m$-dimensional topological manifold with boundary iff for each $x \in X$ there exists $U \subset X$ and homeomorphism $g: U \rightarrow \overline{B_m}$ such that $x \in \operatorname{int}{U}$.

($\overline{B_m}$ denotes the closed unit ball in $\mathbf{R^m}$).

Definition 2. $X$ is called a $m$-dimensional manifold with boundary iff each $x \in X$ has an open neighbourhood which is homeomorphic either with $\mathbf{R^m}$ or with closed halfspace $\mathbf{R}_+^m = \{(x_1,...,x_m) \in \mathbf{R}^m : x_m \geq 0 \}$.

Are the above definitions equivalent (maybe with additional assumption that $X$ is compact metric space) ?

Thanks.

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    @Richard: They look like equivalent definitions to me. I'm not quite sure why the other commenters have an issue with this.2011-09-22

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