mod11lec63 - The space of continuous functions - Part 1

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We begin studying the set of functions Fn(X, Y) between two topological spaces X and Y as a topological space in itself. We consider the topology of pointwise convergence of functions, which is just a reformulation of the product topology. We prove a lemma which justifies this name as well. In case the range space Y is a metric space, we consider another topology, the uniform topology (generated by the uniform metric), which is closely related to the usual notion of uniform convergence of functions. We show that the set of all continuous functions C(X, Y) is closed in Fn(X, Y) in the topology of uniform convergence- this result is usually known as the 'Uniform Limit Theorem'..