I am thinking about see $\mathbb{R}^2$ as one $1-$manifold.
Trying to this I made some considerations.
If I take $\mathbb{R}^2$ with the following topology:
$$(\mathbb{R}^2,\tau) = \text{the product of $\mathbb{R}$ with the the topologies usual and discrete}$$
Precisely:
$$(\mathbb{R}^2,\tau) = (\mathbb{R},\tau_{\text{usual}}) \times (\mathbb{R}, \tau_{discrete})$$
Of course this is a topology. So I was wondering, how to find a differentiable structure on this space such that $(\mathbb{R}^2,\tau)$ is an one manifold?
The problem I am in front is that is hard to find an bijection between this space and $\mathbb{R}$. How to do this? How to choose this in a continuous way?
Thanks!
The result is not a $1$-manifold, since it is not separable - it has no countable dense subset. (Actually, differentiable manifolds are required to be second-countable, which is a significantly stronger requirement than separability.)