How come the definition of analytic continuation doesn't require the smaller and the bigger open subsets to be connected?

Question

The reason that is making me think that these subsets should be connected / simpled connected is because I think that the Taylor disks of convergence of f and F, which is the continuation of f to the bigger subset of the complex plane, should be connected / simply connected.

But Wiki does not even mention connectedness in defining analytic continuation.

Thanks,