Abstract
Topology intervenes in complex variables at various levels. First of all analytic functions are defined in open sets. Connectedness plays a key role in the proof of the uniqueness theorem for analytic functions. The space of functions analytic in an open set is endowed with the topology of uniform convergence on compact sets. This makes this set a metrizable space, and its underlying structure stresses the role of compactness, and plays a key role in the proof of Riemann’s theorem on conformal equivalence of open simply-connected sets (different from C itself) with the open unit disk
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