The Caratheodory Topology for Multiply Connected Domains II

TL;DR

This paper extends the Caratheodory topology to multiply connected domains by defining boundedness, establishing convergence criteria, and generalizing notions of convergence and equicontinuity for families of functions on varying domains.

Contribution

It introduces a new concept of boundedness for families of multiply connected domains and extends convergence and equicontinuity notions within this framework.

Findings

Boundedness ensures the limit domain retains the same connectivity.

Several equivalent conditions for boundedness are established.

The framework allows for generalized convergence and equicontinuity in varying domains.

Abstract

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.