Riemann mapping theorem: Difference between revisions
(New page: ==Statement== Any simply connected domain in <math>\mathbb{C}</math> (i.e. any open, connected and simply connected subset of <math>\mathbb{C}</math>) that is not the whole of <math>\...) |
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Any [[simply connected domain]] in <math>\mathbb{C}</math> (i.e. any open, connected and simply connected subset of <math>\mathbb{C}</math>) that is not the whole of <math>\mathbb{C}</math>, is [[biholomorphically equivalent domains|bihomolorphically equivalent]] to the open unit disc. | Any [[simply connected domain]] in <math>\mathbb{C}</math> (i.e. any open, connected and simply connected subset of <math>\mathbb{C}</math>) that is not the whole of <math>\mathbb{C}</math>, is [[biholomorphically equivalent domains|bihomolorphically equivalent]] to the open unit disc. | ||
==Corollaries== | |||
Any two simply connected domains are homeomorphic (follows from the Riemann mapping theorem and the fact that the unit disc is homeomorphic to <math>\mathbb{C}</math>). | |||
Revision as of 15:19, 20 April 2008
Statement
Any simply connected domain in (i.e. any open, connected and simply connected subset of ) that is not the whole of , is bihomolorphically equivalent to the open unit disc.
Corollaries
Any two simply connected domains are homeomorphic (follows from the Riemann mapping theorem and the fact that the unit disc is homeomorphic to ).