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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 in Hirsch's Differential Topology for a basic discussion of topologies on spaces of differentiable maps as well as Chapter 7 in Kelley's General Topology. – t.b. Sep 19 '11 at 10:59

If you handle some category theory you can also figure out "why" it is good to give that topology to a space of mappings: the kelleyfication of the space $c.o.(X,Y)$ of maps with the compact open topology turns out to be a functor ($c.o.(X,-)^\text{Ke}\colon A\mapsto c.o.(X,A)^\text{Ke}$) which is the best candidate to be a right adjoint to the functor $B\mapsto B\times Y$.

See Mac Lane, Categories for the working mathematician, pp. 185-188, until the end of Thm 3.

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Another important topology on ${\cal C}(X,Y)$ is the pointwise convergence topology, defined as the one having as subbasis sets of the form

$$ S(x,U) = \left\{ f \in {\cal C}(X,Y) \ \ \vert \ \ f(x) \in U \right\} $$

for all points $x\in X$ and all open sets $U\subset Y$.

Some features of this topology:

  1. An open neighbourhood of $f$ with this topology consists of all functions $g$ that are "near" $f$ in just a finite number of points.
  2. This pointwise convergence topology can also be considered on the set $Y^X$, of all maps from $X$ to $Y$ (not just the continuous ones, ${\cal C}(X,Y)$). Or, which is the same, the product of a copy $Y_x = Y$ for each point $x \in X$: $Y^X = \prod_{x\in X} Y_x$. On $Y^X$, this topology agrees with the usual product topology.
  3. And the reason for its name is that a sequence of maps $f_n : X \longrightarrow Y$ converges to $f:X \longrightarrow Y$ with the pointwise convergence topology if and only if, for each $x\in X$, the sequence of points $f_n(x) \in Y$ converges to the point $f(x)\in Y$. (In the same vein, the compact-open topology gives you the uniform convergence.)

You can find all this stuff in Munkress' "Topology", chapter 7.46.

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Observe that the topology of pointwise convergence is the compact open topology with respect to the discrete topology on $X$. – Stefan Geschke Sep 19 '11 at 8:57

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