§ Motivation for the compact-open topology
- If is a compact space and is a metric space, consider two functions .
- We can define a distance .
- The has a maximum because is compact.
- Thus this is a real metric on the function space .
- Now suppose is no longer a metric space, but is Haussdorf. Can we still define a topology on ?
- Let be compact, and let be open such that .
- Since is Hausdorff,