Zero sets of complex analytic functions in more than one variable are never discrete. This can be proved by Hartogs's extension theorem. Domains of holomorphy for single-valued functions consist of arbitrary (connected) open sets. In several complex variables, however, only some connected open sets are domains of holomorphy. The characterization of domains of holomorphy leads to the notion of pseudoconvexity.
Zero sets of complex analytic functions in more than one variable are never discrete. This can be proved by Hartogs's extension theorem. Domains of holomorphy for single-valued functions consist of arbitrary (con,nected) open sets. In several complex variables, however only some connected open sets are domains of holomorphy. The characterization of domains of holomorphy leads to the notion of pseudoconvexity.