In general, let $X$ be a set, $\mathcal N:X \to \wp(\wp(X))$ assign neighbourhood system to every point, $B$ be a filter basis on $\wp(X)$, then we say $B$ converges to $x$, or $B \to x$ iff
$\forall U \in \mathcal{N}_x \exists F \in B(F \subseteq U)$ in which $x \in X$, $\mathcal{N}_x$ is the neighbourhood system of $x$.
However, the pointwise convergence of function is different, let $D$ be a filter basis on $\wp(Y^X)$, $D$ pointwise converges to $f$ iff $Dx \to f(x)$ for all $x \in X$ in witch $f\in Y^X$, $Dx=\{Gx|G \in D\}$, in which $Gx=\{g(x)|g \in G\}$.
My question is can we take the pointwise convergence of function also a convergence of point? Since functions can also be seen as points.
My idea is find a definition of $\mathcal N_f$ such that make these two formula equivalent:
(1)$\forall V \in \mathcal N_f\exists G \in D(G \subseteq V)$
(2)$\forall x \in X\forall U \in \mathcal N_{f(x)}\exists G \in D(Gx \subseteq U)$