Hausdorffness and Regularity of Compact-Open C(X,Y) (7.46.6)

James Munkres Topology, chapter 7.46, exercise 6:

Let $\mathcal{C}(X,Y)$ be under the compact-open topology. Show $\mathcal{C}(X,Y)$ is Hausdorff if $Y$ is Hausdorff, and regular if $Y$ is regular.

Proof: Hausdorffness is simple, as $\mathcal{C}(X,Y)$ inherits a topology at least as fine as that of a subspace under the product topology of $Y^X$, which is Hausdorff when $Y$ is. So suppose $Y$ is regular, let $f∈\mathcal{C}(X,Y)$, and let $K⊆\mathcal{C}(X,Y)$ be a closed subset not containing $f$. Then there exists compact $C_1,...,C_n⊆X$ and open $U_1,...,U_n⊆Y$ such that $f(C_i)⊆U_i$ and for all $g∈K$ there exists $i_g$ such that $g(C_{i_g})⊈U_{i_g}$. Since $f(C_i)$ is compact for each $i$, by regularity of $Y$ choose neighborhoods $V_i$ of these sets such that $\overline{V_i}⊆U_i$. Then $∩B(C_i,V_i)$ is a hood of $f$, and since $$\overline{B(C_i,V_i)}⊆B(C_i,\overline{V_i})$$ we observe $$\overline{∩B(C_i,V_i)}⊆∩\overline{B(C_i,V_i)}⊆∩B(C_i,U_i)$$ is disjoint from $K$.$~\square$