Let $M$ be the subset $[0,1)$ $∪ $ {$2$} of the real line. Find its topological boundary $\mathrm{bd}(M)$ and its manifold boundary $\partial M$.

I know that to find the topological boundary, I need to first find $M$ and $M^c$ and then taking some intersections give me the boundary. But I could not do this.

Also I could not find the manifold boundary. Can anybody please explain the ideas and approach to me clearly?

I know the answers - I guess that the topological boundary $\mathrm{bd}(M)$ is $\{0,1,2\}$ and the manifold boundary $\partial M$ is $\{0\}$.

But I don't know how to show this.


What is the closure of an open interval $(0,1)$? What is the closure of $[0,1)$? What is the interior of each of these? Similarly for $\{2\}$.

For the manifold part, which are the points in $M$ which don't have an open line segment containing them? (Personally I'd say $M$ wasn't a valid manifold with boundary because the $\{2\}$ doesn't have a neighborhood with any structure like an open ball/half-ball.)

  • $\begingroup$ Closure of $[0,1)$ is [0,1] and the same for $(0,1). I have already understood topological boundary by myself. But I dont understand what you say for manifold topology. Please can you expand this more clear? To find manifold boundary is so complecated for me:( thank you for help. $\endgroup$ – B11b May 11 '13 at 8:17
  • $\begingroup$ By the way, $0$ does not have an open line segment. $\endgroup$ – B11b May 11 '13 at 8:27
  • $\begingroup$ Okay I understand now. Fist of all, I find the topological boundary from my theorem. Then, while I find the manifold boundary, I find which points are not on the open line segment. These points are manifold boundary. Right? $\endgroup$ – B11b May 11 '13 at 8:30
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    $\begingroup$ Yes, 0 does not have an open line segment; this is what makes it the manifold boundary. Don't think about the topological boundary whilst you're doing this. More technically, the idea is that locally it looks like the boundary of a half-line, $[0,\infty)$. The points which are analogous to $0$ are the points which are the boundary. For example, $[-7,9]$ has both $-7,9$ as boundaries. $\endgroup$ – Sharkos May 11 '13 at 8:52
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    $\begingroup$ @SeleneAuckland See my answer to your question: math.stackexchange.com/questions/3146131/… $\endgroup$ – Sharkos Mar 13 '19 at 10:52

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