Pushforward filter ($f_{\ast }\mathcal{F}=\{B:f^{-1}(B)\in \mathcal{F}\}$)

For a function $f:X\to Y$, the pushforward of a filter on a set $X$ consists of the subsets $B\subseteq Y$ with $f^{-1}(B)\in \mathcal{F}$. Preimages preserve inclusions and finite intersections, so this is a proper filter. Equivalently it is generated by $f(A)$ for $A\in \mathcal{F}$. Because preimages preserve complements too, the pushforward of an ultrafilter is an ultrafilter.

A point $b$ lies in $\overline{f[A]}$ exactly when some ultrafilter containing $A$ has pushforward converging to $b$. For the forward implication, the sets $A\cap f^{-1}[V]$ for neighborhoods $V$ of $b$ form a proper filter topological basis; extend to an ultrafilter. For the reverse implication each such intersection belongs to a proper filter and is nonempty.