  We continue our exposition concerning the Caratheodory topology for multiply
connected domains by introducing the notion of boundedness for a family of
pointed domains of the same connectivity. The limit of a convergent sequence of
n-connected domains which is bounded in this sense is again n-connected and
will satisfy the same bounds. We prove a result which establishes several
equivalent conditions for boundedness. This allows us to extend the notions of
convergence and equicontinuity to families of functions defined on varying
domains.
