Domains of Holomorphy

We give a simple proof that the notions of Domain of Holomorphy and Weak Domain of Holomorphy are equivalent. This proof is based on a combination of Baire’s Category Theorey and Montel’s Theorem. We also obtain generalizations by demanding that the non-extentable functions belong to a particular class of functions \(X=X({\varOmega })\subset H({\varOmega })\). We show that the set of non-extendable functions not only contains a \(G_{\delta }\)-dense subset of \(X({\varOmega })\), but it is itself a \(G_{\delta }\)-dense set. We give an example of a domain in \(\mathbb {C}\) which is a \(H({\varOmega })\)-domain of holomorphy but not a \(A({\varOmega })\)-domain of holomorphy.