Definition:Compact-Open Topology

Definition

Let $X$ and $Y$ be topological spaces.

Let $\map \CC {X, Y}$ be the set of continuous maps from $X$ to $Y$.

For all compact subsets $K \subset X$ and all open subsets $U \subset Y$, let:

$\map V {K, U} = \set {f \in \map \CC {X, Y}: f \sqbrk K \subset U}$

Let:

$\BB = \set {\map V {K, U}: K \subset X \text{ compact}, U \subset Y \text{ open} }$

The compact-open topology on $\map \CC {X, Y}$ is the topology generated by $\BB$.

Also see

• Compact-Open Topology is Topology which proves that this is in fact a topology.

Sources

• 2000: James R. Munkres: Topology (2nd ed.): $8$: Complete Metric Spaces and Function Spaces: $\S 46$: Pointwise and Compact Convergence