I recently began to study Riemann surfaces, and I got a problem while checking some examples in the book. It is easy to see, for example, that the subset $\{(z, w)\in \mathbb{C}^2\mid w^{2} = \sin z\}$ has a natural complex structure via holomorphic implicit function theorem, so it is a 1-dimensional complex manifold. However, the book does not mention about its connectedness (even though it defines Riemann surfaces as a 1-dimensional connected complex manifold).
I tried to prove its connectedness mainly focusing on the possibility of creating paths between points on it, but due to the problem of selecting branch, it seems quite complicated to me. Can I get any advices for this connectedness problem?
Let $S=\{(z, w)\in \mathbb{C}^2\mid w^{2} = \sin z\}$ be the set in question. The multivalued analytic function $f(z)=\sqrt{\sin z}$ can be continued along any path in $\mathbb C$ that does not pass through the integer multiples of $\pi $.
For every $(z,w)\in S$ with $w\ne 0$ and for every $n\in\mathbb Z$ there is a curve $\gamma$ in the complex plane that goes from $z$ to $\pi n$ and does not meet any other integer multiple of $\pi$. This curve lifts to $S$ via $t\mapsto (\gamma(t), \sqrt{\sin \gamma(t)})$. Note that it's continuous at the endpoint $(\pi n,0)$ as well.
Thus, all $(z,w)\in S$ with $w\ne 0$ and all points of the form $(\pi n,0)$ lie in the same path component of $S$... but they are all of $S$, so the set is path connected.