Banach analytic structures on metrizable topological spaces and a cancellation problem

Abstract

Let X be a topological space whose topology may be defined by a complete metric. Douady (case X compact) and Pestov (the general case) proved that X may be equipped with a structure of Banach analytic set. Here we prove that if X is not discrete, then this may be done in uncountably many ways. Let ( Y, d ) be a non-complete metric space and its completion. Here we prove that we may see as a closed analytic subset of a Banach space in such a way that with V a dense linear subspace of . Then we will prove (using bounded holomorphic functions) the following result. Let X be a connected open subset of an infinite-dimensional locally convex topological vector space V with the weak topology and A an open domain of such that has positive capacity. Then X × A is not biholomorphic to X.