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Why aren't the topological boundary of a manifold with boundary and the boundary of the manifold the same sets?

We defined the boundary of a manifold as follows:

$\partial M:= \lbrace p \in M : \exists (U,\varphi) \text{chart with} \varphi (p)=(0,...,x_n)^T \rbrace$

In contrast to the topological boundary being the closure minus the interior of M.

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  • $\begingroup$ Well, these are different definitions. If you have different definitions, your expectation should be that you might get different results. So your question should read: "why would they be the same"? One of the most common mistakes of beginners in math is to look at the name and draw conclusions from associations resulting from that name. It's the definition you have to look at. $\endgroup$ – Thomas Oct 4 '18 at 10:50
  • $\begingroup$ Ok so it is caused by the different nature of the definition, thank you. $\endgroup$ – Rico1990 Oct 4 '18 at 11:48
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Take a simple example that lives in both situations. If $X$ is any topological space, the topological boundary of $X$ is $\varnothing$. Now, consider the case $X=[0,1]$ endowed with the inherited Euclidean subspace topology. This is also the simplest example of a $1$-manifold with boundary. As I said earlier, the topological boundary is $\varnothing$. However, the manifold boundary is the $2$-point set $\{0,1\}$. If you instead regard $[0,1]$ as a subset of the space $\mathbb{R}$ then its topological boundary is $\{0,1\}$, which is where your confusion lies.

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    $\begingroup$ Thank you very much for that example. $\endgroup$ – Rico1990 Oct 4 '18 at 16:34

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