Show that if \(Y\) is locally compact Hausdorff, then composition of maps
\[ \mathcal{C}(X, Y) \times \mathcal{C}(Y, Z) \to \mathcal{C}(X, Z) \]
is continuous, provided the compact-open topology is used throughout. [Hint: If \(g \circ f \in S(C, U)\), find \(V\) such that \(f(C) \subset V\) and \(g(\overline{V}) \subset U\).]
Solution
For compact \(C\) and open \(U\), write \(S(C, U) = \{h : h(C) \subseteq U\}\); these sets form a subbasis of the compact-open topology. It suffices to show that every point \((f, g)\) of the preimage of a subbasic set \(S(C, U) \subseteq \mathcal C(X, Z)\) under composition has a neighborhood inside that preimage.
1. If \(g(f(C)) \subseteq U\), there is an open \(V \subseteq Y\) with \(f(C) \subseteq V\), \(\overline{V}\) compact, and \(g(\overline{V}) \subseteq U\).
The set \(g^{-1}(U)\) is open and contains the compact set \(f(C)\). Since \(Y\) is locally compact Hausdorff, each \(y \in f(C)\) has an open neighborhood \(W_y\) with \(\overline{W_y}\) compact and \(\overline{W_y} \subseteq g^{-1}(U)\). Finitely many \(W_{y_1}, \ldots, W_{y_k}\) cover \(f(C)\); put \(V = W_{y_1} \cup \cdots \cup W_{y_k}\). Then \(\overline{V} = \overline{W_{y_1}} \cup \cdots \cup \overline{W_{y_k}}\) is compact and contained in \(g^{-1}(U)\).
2. \(S(C, V) \times S(\overline{V}, U)\) is an open neighborhood of \((f, g)\) whose image under composition lies in \(S(C, U)\).
By step 1, \(f(C) \subseteq V\) and \(g(\overline{V}) \subseteq U\), and \(\overline{V}\) is compact, so \((f, g) \in S(C, V) \times S(\overline{V}, U)\), a product of subbasic open sets. If \(f' \in S(C, V)\) and \(g' \in S(\overline{V}, U)\), then \(g'(f'(C)) \subseteq g'(\overline{V}) \subseteq U\).
3. Q.E.D.
By step 2, the preimage of every subbasic open set of \(\mathcal C(X, Z)\) is open, so composition is continuous.