Exercise 18, § 8

Let $X_1, X_2$ be two sets, let $\mathfrak{F}_i$ be a filter on $X_i$ ($i = 1, 2$), and let $f$ be a mapping of $X_1 \times X_2$ into a regular space $Y$ such that, for each $x_1 \in X_1$ there exists $\lim_{x_2, \mathfrak{F}_2} f(x_1, x_2) = g(x_1)$.

Show that a point $a \in Y$ is a cluster point of $g$ with respect to the filter $\mathfrak{F}_1$ if and only if, given any neighbourhood $V$ of $a$ in $Y$ and any set $A_1 \in \mathfrak{F}_1$, there exists $x_1 \in A_1$ and $A_2 \in \mathfrak{F}_2$ such that $f(\{x_1\} \times A_2) \subset V$.