Exercise 2.02

By clever choice of coordinate charts, can we make $\mathbb{R}^{2}$ look like a one-dimensional manifold? Can we make $\mathbb{R}^{1}$ look like a two-dimensional manifold? If so, explicitly construct an appropriate atlas, and if not, explain why not. The point of this problem is to provoke you to think deeply about what a manifold is; it can't be answered rigorously without going into more details about topological spaces. In particular, you might have to forget that you already know a definition of "open set" in the original $\mathbb{R}^{2}$ or $\mathbb{R}^{1}$, and define them as being appropriately inherited from the $\mathbb{R}^{1}$ or $\mathbb{R}^{2}$ to which they are being mapped.

We can go back and forth from $\mathbb{R}^{1}$ to $\mathbb{R}^{2}$ using the following map, 

• from $\mathbb{R}^{1}$ to $\mathbb{R}^{2}$: take every second digit (base 10 for example) starting from the right of the decimal point for the first coordinate and every second digit starting from its left for the other, 

• from $\mathbb{R}^{2}$ to $\mathbb{R}^{1}$: sew the digits of the two coordinates together in the reverse operation.