I asked the same question on MathStackExchange a month ago and received no answer. I feel that this would be more suitable for MathOverflow.

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 $f: X\rightharpoonup Y$ 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?