Deciding whether a given set is a manifold This question should be extremely elementary but it stumped me somewhat. I know the definition of a (topological/smooth) manifold but I seem to have trouble when it comes to deciding whether a given subset of $\mathbb{R}^n$ is a manifold. In particular I was recently asked to say whether the following were manifolds:


*

*The set $M_1 = \{(x, y, z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 = 1, z \geq 0\}$

*The set $M_2 = \{(x, y) \in \mathbb{R}^2 : |x| + |y| = 1\}$


My intuition is that  because the set $M_1$ has a "boundary" (the unit circle in the $xy$ plane) it isn't a manifold, but I'm not really sure of how to say taht precisely. In general I'm not exactly sure why having a boundary is a problem. On the other hand the set $M_2$ is just a square (I think), which is topologically the same as $S^1$, and so it should be a manifold since $S^1$ is. But I thought manifolds weren't supposed to have "sharp corners"? At least I remember hearing words to that effect. (To be honest I don't know what the precise definition of a "sharp corner" is).
Can anyone help me with this? Obviously I'm stilled a bit unclear about the concept of a manifold (even though I guess I understand the definition).

$M_1$ is what is known as a "manifold with boundary". If you haven't been introduced to those, yet, then the desired answer is probably that $M_1$ isn't a manifold at all.
You're right that $M_2$ is a square. This is then topologically equivalent to $S^1$, so is a topological manifold. However, it is not a smooth manifold, precisely because of those "sharp" corners.
