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According to the definition of a manifold with boundary, there are two types of coordinate charts of a $n$-dimensional manifold with boundary: interior charts and boundary charts. A boundary chart is a homeomorphism $\phi: U \rightarrow V$ where $U$ is an open subset containing a boundary point of $M$ while $V$ is an open subset of the half space $H^n = \{(x_1, \ldots, x_n) \in \mathbb{R}^n: x_n \geq 0\}$ such that $V \cap \partial H^n \not= \emptyset$.

Now according to topology, an open subset of $H^n$ is a set $V$ such that $V = O \cap H^n$ where $O$ is an open subset of $\mathbb{R}^n$. This is called induced topology because we have $H^n \subset \mathbb{R}^n$.

But I cannot understand how one defines an open subset containing a boundary point of $M$ as $M$ is not supposed to be a subset of any ambient space.

Ka Fat Chow
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  • To define a topology on M, it suffices to describe a basis of open sets formed using the coordinate charts. Then an arbitrary open set is a union of basis elements – Alex Ortiz Aug 08 '22 at 03:18
  • In this way, isn't the topology of $M$ depending on the coordinate charts? – Ka Fat Chow Aug 08 '22 at 03:31

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