Proposition 7

A point $y \in Y$ is a limit of $f$ with respect to the filter $\mathfrak{F}$ if and only if, for each neighbourhood $V$ of $y$ in $Y$, there is a set $M \in \mathfrak{F}$ such that $f(M) \subset V$ (i.e. $\overline{f}(V) \in \mathfrak{F}$ for each neighbourhood $V$ of $y$). A point $y \in Y$ is a cluster point of $f$ with respect to $\mathfrak{F}$ if and only if for each neighbourhood $V$ of $y$ and each $M \in \mathfrak{F}$ there is a point $x \in M$ such that $f(x) \in V$.