### §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 *C*^{0}-**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 ...