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Looking at Wikipedia, I see the Riemann surface for the log, square root, and other multi-valued functions are all connected and are constructed via the inverse of an analytic function.

It seems like connectivity is a necessity, but I can't figure out why or if it's even true.

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  • $\begingroup$ Riemann Surfaces are smooth manifolds, and are thus path connected and thus connected. Note they are rarely simply connected, e.g. the Riemann Surface for the logarithm (this is important, as this can be seen to be where branches come from) $\endgroup$ – Brevan Ellefsen May 2 '19 at 0:54
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    $\begingroup$ ^ Smooth manifolds don't have to be connected by all definitions I have seen. But Riemann surfaces of analytic functions are connected by definition. $\endgroup$ – Jane Doé May 2 '19 at 0:57
  • $\begingroup$ @JaneDoé Not all definitions of manifold require connectedness, but I am unfamiliar with a definition of smooth manifold that does not require it... you can't get very far with smooth manifolds without needing second countability and Hausdorffness, e.g. partitions of unity, Riemannian metrics, etc. Smoothness is a strong condition. Regardless, Riemann Surfaces are by definition complex manifolds and fall under the second countable + Hausdorff definition. $\endgroup$ – Brevan Ellefsen May 2 '19 at 1:00
  • $\begingroup$ @JaneDoé What definition do you refer to? $\endgroup$ – Shane P Kelly May 2 '19 at 1:01
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    $\begingroup$ I don't think it's standard to assume that even smooth ($C^\infty$) manifolds are connected. This is not necessary to define a smooth structure, so why assume it? For instance, this textbook does not assume it: webmath2.unito.it/paginepersonali/sergio.console/lee.pdf . As for Riemann surfaces, the definition on wikipedia, for instance, requires that they be connected. And it makes sense intuitively if we think of patching together charts on which the function in question is single-valued. These charts will have overlaps are are path-connected, so the R.S. should also be connected. $\endgroup$ – Jane Doé May 2 '19 at 1:08

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