I asked [the same question](https://math.stackexchange.com/questions/4802488/compact-open-topology-for-partial-maps) on MathStackExchange a month ago and received no answer. I feel that this would be more suitable for MathOverflow. 


[Compact open topology](https://mathoverflow.net/questions/130287/compact-open-topology) is one of the most common ways of turning the set of continuous maps between two topological spaces into a topological space. Suppose $X, Y$ are topological spaces, and let $K\subseteq X$ be a compact subset and $U\subseteq Y$ be an open subset. $$V_{K, U}=\{f: X\to Y \mid f(K)\subseteq U\}$$ Then the collection of all such $V_{K, U}$ is a subbase for the compact-open topology on $C(X, Y).$

A [partial map](https://en.wikipedia.org/wiki/Partial_function#Function_spaces) $f: X\rightharpoonup Y$ is a function from some subset $X_f\subseteq X$ to $Y.$ To define continuity  in this context, consider the totalization 
$$\widetilde{f}(x) =
\begin{cases}
f(x),  & \text{if $x\in X_f$} \\
\star, & \text{if $x\notin X_f$}
\end{cases}$$ where $\star\notin Y,$ and equip $\widetilde{Y}=Y\cup\{\star\}$ with the smallest topology extending that of $Y.$ i.e., $U\subseteq \widetilde{Y}$ is open iff $U=\widetilde{Y}$ or $U$ is open in $Y.$ Then claim $f: X\rightharpoonup Y$ is continuous iff $\widetilde{f}: X\to\widetilde{Y}$ is continuous. Observe that this definition doesn't alter the continuity of total maps. 


Now, I want to understand how we can have a nice topology on the set of all continuous partial maps between $X$ and $Y.$ Since there is a bijection between $\{X\rightharpoonup Y\}$ and $\{X\to \widetilde{Y}\},$ I'm thinking that we can simply look at the compact-open topology on $C(X, \widetilde{Y}).$ But I'm not sure that this is the correct/best way to topologize the space of partial maps. I need the opinion of somebody who has some experience in this area. 

 
- Are there other ways of defining continuity for partial maps?
- If so, how is this topology different from those?
- Has anyone studied this problem before?