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Given two topological spaces $X, Y$, the only example I know of a topology on the space $\mathcal C(X,Y)$ of continuous mappings from $X$ to $Y$ is the compact-open topology. However I presume that there are other interesting topologies as well, which are useful in other situations. What are some examples, and what is a most interesting situation for its use?

In particular, is there any particular interesting topology if $X$, $Y$ are both smooth manifolds and we are considering differentiable maps instead of continuous maps?

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    You may want to consult [Chapter 2](http://books.google.com/books?id=iSvnvOodWl8C&pg=PA34) in Hirsch's *Differential Topology* for a basic discussion of topologies on spaces of differentiable maps as well as [Chapter 7](http://books.google.com/books?id=-goleb9Ov3oC&pg=PA217) in Kelley's *General Topology*.2011-09-19

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