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Dynamics in One Complex Variable. (AM-160), 3rd Edition by John Milnor

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§3. Normal Families: Montel’s Theorem

Let S and T be Riemann surfaces. This section will study compactness in the function space Hol(S, T) consisting of all holomorphic maps with source S and target T. We first define a topology on this space, and on the larger space Map(S, T) consisting of all continuous maps from S to T. This topology is known to complex analysts as the topology of uniform convergence on compact subsets, or more succinctly as the topology of locally uniform convergence. It is known to topologists as the compact-open topology (Problem 3-a), or when dealing with smooth manifolds as the C0-topology.

Definition. Let X be a locally compact space and let Y be a metric space. For any f in the space Map (X, Y) of continuous maps from ...

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