We continue our exposition concerning the Carathéodory topology for multiply connected domains which we began in [Comerford M., The Carathéodory topology for multiply connected domains I, Cent. Eur. J. Math., 2013, 11(2), 322–340] 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.
@article{bwmeta1.element.doi-10_2478_s11533-013-0365-y,
author = {Mark Comerford},
title = {The Carath\'eodory topology for multiply connected domains II},
journal = {Open Mathematics},
volume = {12},
year = {2014},
pages = {721-741},
zbl = {1297.30045},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0365-y}
}
Mark Comerford. The Carathéodory topology for multiply connected domains II. Open Mathematics, Tome 12 (2014) pp. 721-741. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0365-y/
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