Topologies defined by some invariant pseudodistances

Abstract

Complex analysis deals with holomorphic mappings between complex spaces. We ought to allow complex spaces to be infinite-dimensional and to have singularities. There are well-developed theories in the finite-dimensional case (see e.g. [GR], [Lo]) and for domains in certain kinds of topological vector spaces (see e.g. [Di], [FV], [He]). The most obvious approach to infinite-dimensional complex spaces is too general to yield sensible results [Do], but there has been progress [Au] in the study of (possibly infinite-dimensional, possibly singular) semi-Fredholm-analytic spaces. In any case, I shall assume that a complex space carries the structure of a Hausdorff space, which I shall call its underlying topology ; in the finite-dimensional case I shall assume that the underlying topology is paracompact. This note discusses some foundational questions and summarizes what we know about the relationships between the underlying topology and the topologies induced by the classical pseudodistances of Caratheodory and Kobayashi. Let Hol(X,Y ) denote the set of all holomorphic mappings from the complex space X into the complex space Y ; I shall assume that these mappings are continuous with respect to the underlying topologies of X and Y . Let D denote the open unit disc in C. A mapping f ∈ Hol(D, X) is called an analytic disc in X. I shall also assume that a complex space has the property that any two points can be connected by (the images of) a finite chain of analytic discs in the space; in the finite-dimensional case this is equivalent to connectedness in the topological sense [K70, pp. 97–98], [La, pp. 15–16].