ABSTRACT
To any open subset D ⊆ ℂ n there is canonically associated its envelope of holomorphy E(D), the maximal holomorphic extension of D in the category of Riemann domains for which holomorphic functions separate points. If E(D) is also an open subset of ℂ n , then it is clear from Theorem G8 that E(D) is a domain of holomorphy. The envelopes of holomorphy of subsets in ℂ n , or more generally the envelopes of holomorphy of Riemann domains for which holomorphic functions separate points, are thus natural candidates to be considered the analogues for Riemann domains of domains of holomorphy. From this point of view it is fairly natural to limit the consideration to those Riemann domains for which holomorphic functions are assumed to separate points, as will be done whenever convenient in this section. The extent to which such an assumption is necessary will be discussed in a later section.
