Infinitely-connected domain

domain of infinite connectedness

A domain for which the 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 [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

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