Q1: Let $U$ be an open subset of $\mathbb{R}^n$. Is this a smooth manifold?

A1: Yes, an atlas is given by one chart and can be extended to a smooth structure.

A2: No, there is no way to define an atlas for $U$.

Q2: Is the map $\omega: [0,1] \rightarrow \mathbb{R}$ given by $$ \omega(x) = \sqrt{1-x}$$ a $C^1$-diffeomorphism?

A1: No, it’s not differentiable at $0$.

A2: No, it’s not differentiable at $1$.

A3: Yes, it’s differentiable everywhere and bijective.

A4: Yes, it’s differentiable everywhere, bijective and the inverse is also differentiable.

Q3: Is the map $\omega: (0,1) \rightarrow \mathbb{R}$ given by $$ \omega(x) = \sqrt{1-x}$$ a $C^1$-diffeomorphism?

A1: No, it’s not differentiable at $\frac{1}{2}$.

A2: No, it’s not differentiable at all points.

A3: Yes, it’s differentiable everywhere and bijective.

A4: Yes, it’s differentiable everywhere, bijective and the inverse is also differentiable.

Q4: Is the sphere $S^n$ with the atlas given in previous videos a $C^\infty$-smooth manifold?

A1: Yes, it’s a $C^\infty$-smooth atlas which can be extended to a maximal one.

A2: No, it’s not a $C^\infty$-smooth atlas!

A3: No, it’s a $C^\infty$-smooth atlas but it cannot be extended to a maximal one.