Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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.

share|cite|improve this answer

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.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.