Topological Spaces

Abstract

Topological and metric spaces permeate complex analysis in numerous places. For instance, the space $\mathcal{H}(\Omega)$ of functions analytic in some open domain $\Omega$ is a Frechet space (and moreover, a Montel space). Hilbert and Banach spaces of analytic functions also play an important role. These examples are in fact instances of topological vector spaces. As another example, Riemann surfaces are in particular analytic manifolds, and as such are special cases of topological spaces, and in fact of metric spaces.

Les premiers faits topologiques rencontres en theorie des fonctions analytiques ont apparu avec l’introduction des surfaces de Riemann.

S. Stoilow, Lecons sur les principes topologiques de la theorie des fonctions analytiques, [299, Preface, p. v].