Compact-open topology

Remarks

If $(Y, d)$ is a metric space, the compact-open topology is the topology of uniform convergence on compact subsets of $X$. In particular, a sequence $(f_n)$ converges to $f$ if and only if $\sup_{x \in K} d(f_n(x), f(x)) \to 0$ for every compact $K \subseteq X$. If $X$ is locally compact Hausdorff, the evaluation map $C(X, Y) \times X \to Y$, $(f, x) \mapsto f(x)$, is continuous.

Example: for $X = [0, 1]$ and $Y = \R$, the space $X$ is compact, so convergence in $C([0, 1], \R)$ is uniform convergence on $[0, 1]$.

Non-example: the topology of pointwise convergence on $C([0, 1], \R)$ is not the compact-open topology. Let $f_n$ be the piecewise linear map with $f_n(0) = 0$, $f_n(1/n) = 1$ and $f_n(x) = 0$ for $x \ge 2/n$. Then $f_n(x) \to 0$ for each $x$, but $f_n \notin V([0, 1], (-1, 1))$ for every $n$, so $f_n$ does not converge to $0$ in the compact-open topology.