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I saw that a $m-$manifold with boundary is defined by the following. Every point of $M$ has a neighborhood homomorphic to either an open ball in $\mathbb{R}^m$ or the upper half of $\mathbb{R}^m$ which is known as $\mathbb{H}^m$.

But how do I intuitively realise in the first impression if a manifold given is with boundary or without boundary.

For example which of the following are manifolds with/without boundary?

1-Sphere $S^n$

2-$n-$ball $D^n$.

3-Torus

4-$\mathbb{R}P^n$

5-$\mathbb{C}P^n$

6-Orientable surface of genus $g$

7-Open Mobius Strip

8- Closed Mobius Strip

9-Klein Bottle

permutation_matrix
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    Intuitively if you „live“ on a manifold without boundary everything around you looks like $\mathbb R^m$, so you see the horizon with no end, e.g. on $S^n$ or the torus. The space around boundary points look like $\mathbb H^m$, which means beings in this universe see a wall in one direction in the horizon, e.g on the closed disk $\bar D^m$. The boundary here is the sphere $S^m$. The rest is not easy to view intuitively since you lose the intuition offered by the Euclidean space in which the sphere, the torus and the ball sit in. – T.P. Dec 04 '22 at 15:07
  • @T.P., $(0,1)$ is a smooth manifold without boundary and $[0,1]$ is a smooth manifold with boundary? – permutation_matrix Dec 14 '22 at 18:15
  • Yes, that’s right. And the boundary of $[0,1]$ consists of two points, namely ${0,1}$. – T.P. Dec 14 '22 at 18:20

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