Locally analytically connected topologies

Abstract

We shall prove the existence of two natural analytic topologies which refine the given topology on a space with a specified collection of continuous complex-valued functions. Whereas one would expect [2] the two constructions always, to yield the same analytic refinement, we show by example that is not the case.

This work was motivated by the search for analytic structure in the maximal ideal space of a function algebra, in particular the existence of nontrivial holomorphic mappings of analytic varieties into the maximal ideal space. (For restrictions on what type of analytic varieties need be considered, see Section 4.)