The Caratheodory Topology for Multiply Connected Domains II
  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. 
Mark Comerford
math.CV