Understanding topological and manifold boundaries on the real line

Question

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.

Answer 1

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.)