Is there a natural reason for defining the compact-open topology on the set of continuous functions between two locally compact spaces. For example "to make ... functions continuous". Or in another way of asking this, is there an adjoint functor of the functor, say F, which assigns the topological space $F(X,Y):=Hom\_C(X,Y)$ (with the compact-open topology on it) to the couple X,Y.



compact-open topology
In regard to your question I recommend *[Topologies on spaces of continuous functions,](http://www.cs.bham.ac.uk/~mhe/papers/newyork.pdf) Topology Proceedings, volume 26, number 2, pp. 545-564, 2001-2002* by Martin Escardo and Reinhold Heckmann.


