Specifically, I'm wondering, if X and Y are Hausdorff, and Y is compactly generated, does it follow that C(X,Y), with the compact-open topology, is compactly generated?


Edit: answered as written, but curious about other conditions that do imply the compact-open topology is compactly generated. E.g. if we strengthen the assumption to Y being locally compact?



Under what conditions is the compact-open topology compactly generated?
Not necessarily: consider the compactly generated space $Y=\mathbb R^\infty=\varinjlim \mathbb R^n$, which is the direct limit of Euclidean spaces. Then for the countable discrete space $X=\omega$ the function space $C(X,Y)$ is homeomorphic to $(\mathbb R^\infty)^\omega$ and hence is not sequential and so is not compactly generated. To see that the space $(\mathbb R^\infty)^\omega$ is not sequential, one should apply the known fact that the product $\mathbb R^\infty\times\mathbb R^\omega$ is not sequential.


