We know that the compact open topology on $\mathcal{C}(X,Y)$ is a good choice for topology on the set of continuous maps, but this seems really efficient, both naively and with respect to existence of good theorems, when $X$ is a locally compact Hausdorff topological space.
For instance, given a map $H:A \times X \to Y$, then, if $X$ is locally compact Hausdorff, we can assure that
$H$ is continuous if and only if the induced map $\widetilde{H}: A \to \mathcal{C}(X,Y)$ given by $\widetilde{H}(a)=H(a, \cdot)$ is continuous.
Now, in my intuition, I prefer to see homotopies as the maps $\widetilde{H}$, when $A=[0,1]$. However my point of view is not formally justified, given that there is an inconsistency regarding to what are homotopies in the two points of view, since the "iff" statement above need not hold on general spaces.
Therefore, I was thinking, and this is my question:
Q: What happens when instead of considering the compact open topology, we consider the final topology on $\mathcal{C}(X,Y)$, with respect to the family $\{\widetilde{H_{\lambda}}\}_{\lambda \in \Lambda}$ of induced maps coming from continuous maps $H_{\lambda}: A \times X \to Y$? Is this manageable/interesting/has this been done?
Relevant: I've seen the following questions:
Viewing Homotopies as Paths in $\mathcal{C}^0(X,Y)$ ,
https://mathoverflow.net/questions/35246/the-definition-of-homotopy-in-algebraic-topology/ ,
but none of them addresses the suggestion made here, which is the main interest of this question.