Given a topological subspace M of $\mathbb{R}^2 \times S^1$ defined by $(x,y,e^{i\theta})$ and two charts $(U,h)$, $(V,k)$ such that
$H:\mathbb{R} \times (-\pi,\pi) \to M$
$H(x,\theta) = (x,x\tan(\frac{\theta}{2}),e^{i\theta}) U= \operatorname{im}(H)$, $h= H^{-1}$
$K:\mathbb{R} \times (0,2\pi) \to M$
$K(y,\theta) = (y\cot(\frac{\theta}{2}),y,e^{i\theta}) V= \operatorname{im}(K), k= K^{-1}$
I've shown that $M$ is Hausdorff but how do I prove that it has a countable basis, $M = U\cup V$, and that the transition map and its inverse are differentiable.