postcomposition for the compact open topology

Let \((X, \mathcal{T}_X)\), \((Y, \mathcal{T}_Y)\) and \((Z, \mathcal{T}_Z)\) be topological space and \(f: Y \rightarrow Z\) a continuous map Given the mapping space \(\mathcal{C}(X,Y)\) with the compact open topology, postcomposition with \(f\) is a continuous map

\begin{align*} f^{ \leftarrow} : \mathcal{C}(X,Y) \rightarrow& \mathcal{C}(X,Z) \\ g \mapsto& f \circ g \end{align*}

continuous maps and subbasis

Let \(\mathcal{O}_{K,U} \subseteq \mathcal{C}(X,Y)\) be open. Then the preimage is

\begin{align*} (f^{ \leftarrow})^{-1}(\mathcal{O}_{K,U}) =& (f^{ \leftarrow})^{-1}\{g \in \mathrm{Hom}_{\mathrm{Top}}(X,Z) \vert g[K] \subseteq U\} \\ =& \{g' \in \mathrm{Hom}_{\mathrm{Top}}(X,Y) \vert g'[K] \subseteq f^{-1}[U]\} \end{align*}

where \(f^{-1}[U]\) is open, since \(f\) is continuous, hence

\begin{align*} \{g' \in \mathrm{Hom}_{\mathrm{Top}}(X,Y) \vert g'[K] \subseteq f^{-1}[U]\} = \mathcal{O}_{K, f^{-1}[U]} \end{align*}