Suppose $Y$ is regular.
Let $f^*$ be point in $δ(X,Y)$ and $\{f:f(K)∈G\}$ open set such that $f^*∈\{f:f(K)⊆G\}$.
Since $f^* (K)⊆G$, and ($Y$ regular) then $∀y∈f^* (K), ∃v_y$ open set such that $y∈v_y⊆(v_y ) ̅⊆G$.
Since $f^*$ is continuous, then $f^* (K)$ is compact, and the open covering $\{v_y:y∈f^* (K)\}$ must have a finite subcover. The union of this subcover $V$ will have the property $f^* (K)⊆V⊆V ̅⊆G$
Hence $f^*∈\{f:f(K)⊆V\}$.
If $g∉\{f:f(K)∈\bar{V} \}$, then $∃x∈K∋g(x)∉\bar{V}$.Thus, $π_x^{-1}(X| \bar{V})=\{f:f(x)∈X | \bar{V} \}$ is an open set in the “point-open” topology and hence also in the “compact-open” topology which contains $g$ is disjoint of the form $\{f:f(K)⊆V\}$.
Hence $Cl(\{f:f(K)⊆V\}) ⊆\{f:f(K)∈\bar{V} \}⊆\{f:f(K)∈G\}$.
Let $f^*$ be contained in a general open set $\mathcal{G}^*$ which then contains a basis open set $\{f:f(K_1 )⊆G_1 \}∩…∩\{f:f(K_n )⊆G_n \}⊆\mathcal{G}^*$ where $f^* (K_i )⊆G_i$ that is, a basis open set containing $f^*$. By the above construction, the set $V^*=\{f:f(K_1 )⊆V_1 \}∩…∩\{f:f(K_n )⊆V_n\}$ is an open set such that $f^*∈V^*⊆(V^* ) ̅⊆\mathcal{G}^*$.
I am trying to understand the above proof. I understand most of the proof what we are trying to do but I fail to understand the quoted part and how it is related to the rest of the proof. Could you please explain to me the quoted part. Another presentation of the prrof.
The quoted part just shows that $\{f : f[K] \subseteq \overline{V}\}$ is closed in the compact-open topology. It does this by sowing that its complement is even pointwise-open so in particular compact open. As $\{f: f[K]\subseteq V\}$ is a subset of this closed set, the closure of it is also a subset of $\{f: f[K]\subseteq \overline{V}\}$ and finally that set is itself trivially (by definition) a subset of $\{f: f[K]\subseteq G\}$ as $\overline{V}\subseteq G$. Etc.
So for the subbasic sets of the compact-open topology we can find, for each point, some subbasic open set that contains it and whose closure sits inside that first subbasic open set; it's a first step towards doing this for all open sets (as we need to do; do we reduce this to the base generated by the subbasic sets at the end).