View source for Infinitely-connected domain

''domain of infinite connectedness''

A domain for which the [[Fundamental group|fundamental group]] is not finitely generated. The concept of an infinitely-connected domain is usually employed for domains in the extended complex plane, and in such a case the above definition is equivalent to the fact that the boundary of the domain consists of an infinite number of boundary components. The topological properties of infinitely-connected domains were studied by P.S. Urysohn [[#References|[1]]]. The fundamentals of the theory of analytic and univalent functions in infinitely-connected domains were established mainly by P. Koebe

and H. Grötzsch .

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Urysohn,   "Works on topology and other areas of mathematics" , '''1–2''' , Moscow-Leningrad  (1951)  (In Russian)</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  P. Koebe,   "Ueber die Uniformisierung beliebiger analytischer Kurven"  ''Nachr. Ges. Wiss. Göttingen''  (1909)  pp. 324–361</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  P. Koebe,   "Zur konformen Abbildung unendlich-vielfach zusammenhängender schlichter Bereiche auf Schlitzbereiche"  ''Nachr. Ges. Wiss. Göttingen''  (1918)  pp. 60–71</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top">  H. Grötzsch,   "Zum Parallelschlitztheorem der konformen Abbildung schlichter unendlich-vielfach zusammenhängender Bereiche"  ''Ber. Verh. Sächs. Akad. Wiss. Leipzig Math. Naturwiss. Kl.'' , '''83'''  (1931)  pp. 185–200</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top">  H. Grötzsch,   "Das Kreisbogenschlitztheorem der konformen Abbildung schlichter Bereiche"  ''Ber. Verh. Sächs. Akad. Wiss. Leipzig Math. Naturwiss. Kl.'' , '''83'''  (1931)  pp. 238–253</TD></TR><TR><TD valign="top">[3c]</TD> <TD valign="top">  H. Grötzsch,   "Ueber die Verzerrung bei schlichter konformer Abbildung mehrfach zusammenhängender schlichter Bereiche III"  ''Ber. Verh. Sächs. Akad. Wiss. Leipzig Math. Naturwiss. Kl.'' , '''83'''  (1931)  pp. 283–297</TD></TR><TR><TD valign="top">[3d]</TD> <TD valign="top">  H. Grötzsch,   "Ueber Extremalprobleme bei schlichter konformer Abbildung schlichter Bereiche"  ''Ber. Verh. Sächs. Akad. Wiss. Leipzig Math. Naturwiss. Kl.'' , '''84'''  (1932)  pp. 3–14</TD></TR></table>