compact open topology

Let \((X,\mathcal{T})\) and \((Y,\mathcal{T}_Y)\) be topological spaces.

Then the compact open topology on the mapping space \(\mathcal{C}(X,Y)\) is defined by the Subbasis containing the subsets

\begin{align*} \mathcal{O}_{K, U} = \{f: X \rightarrow Y \vert f \in \mathrm{Hom}_{\mathrm{Top}}(X,Y), f[K] \subseteq U, K \subseteq X \text{ compact}, U \subseteq Y \text{ Open} \} \end{align*}