ABSTRACT
The field of real numbers is denoted by ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq27.tif"/> and the field of complex numbers by ℂ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq28.tif"/> . Both are topological fields with well-known properties. In studying the theory of functions of several complex variables, the space of primary interest is the n-dimensional complex vector space ℂ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq29.tif"/> , which can be identified with the Cartesian product ℂ × ⋯ × ℂ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq30.tif"/> of n copies of the complex plane, or with the ordinary Euclidean space ℝ 2 n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq31.tif"/> of dimension 2n. A point of ℂ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq32.tif"/> is denoted by Z = (z 1,…, z n ), where Zj = Xj + iyj , with Xj and yj real numbers and i a fixed square root of − 1. In considering the topology of ℂ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq33.tif"/> , two special bases for the open subsets are particularly useful. An open polydisc in ℂ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq34.tif"/> is a subset of the form Δ ( A ; R ) = Δ ( a 1 , … , a n ; r 1 , … , r n ) = { Z ∈ C n : | z j − a j | < r j for 1 ≦ j ≦ n } , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq35.tif"/>
