Let $X,Y$ be topological spaces. The compact-open topology on $C(X,Y)$ is generated by the sub-basic open sets $$ \left\{U_{K,O}: \mbox{ K is compact in X and O is open in Y}\right\}\\ U_{K,O}:=\left\{f \in C(X,Y):\, f(K')\subseteq O \right\} $$.$$ \left\{U_{K,O}: \mbox{ K is compact in X and O is open in Y}\right\}\\ U_{K,O}:=\left\{f \in C(X,Y):\, f(K')\subseteq O \right\} .$$ However, if $X$ is not Hausdorff then sometimes this topology may be oddly-behaved.
Instead, consider this generalization (which clearly is equal to the compact-open topology when $X$ is Hausdorff) of the compact-open topology; generated by the sub-basic sets: $$ \left\{V_{K',O}:(\exists K \subseteq X \mbox{ compact}) K'\subseteq K, \mbox{ K is closed in X and O is open in Y}\right\}\\ V_{K',O}:=\left\{f \in C(X,Y):\, f(K')\subseteq O \right\}. $$$$ \left\{V_{K',O}:(\exists K \subseteq X \mbox{ compact})\; K'\subseteq K, \mbox{ K is closed in X and O is open in Y}\right\}\\ V_{K',O}:=\left\{f \in C(X,Y):\, f(K')\subseteq O \right\}. $$
Is this topology studied? If so where?
