The following is claimed in Differential Topology by Hirsch (Theorem 2.4.1, p. 59), with $X,Y$ topological spaces and $C_W(X,Y)$ the compact-open or "weak" topology on the set of continuous maps from $X$ to $Y$:
Theorem. Let each component of $X$ be locally compact with a countable base; let $Y$ be a complete metric space. Then $C_W(X,Y)$ has a complete metric.
The proof begins as follows:
Proof. It suffices to construct a complete metric on $C_W(X_\alpha,Y)$ for each component $X_\alpha$ of $X$; therefore we assume $X$ locally compact with a countable base.
My question: why is this true if $X$ has uncountably many components?
If $X$ had countably many components $X_1,X_2,\ldots$ with complete metrics $d_1,d_2,\ldots$ on $C_W(X_1,Y),C_W(X_2,Y),\ldots$ then I believe I can define a complete metric on $C_W(X,Y)$ by the formula $$d(f,g):= \sum_{i=1}^\infty \frac{1}{2^i}\frac{d_i\left(f|_{X_i},g|_{X_i}\right)}{1+d_i\left(f|_{X_i},g|_{X_i}\right)},$$
but I can't figure out what to do if $X$ does not have countably many components.