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:
- An open neighbourhood of $f$ with this topology consists of all functions $g$ that are "near" $f$ in just a finite number of points.
- 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.
- 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.