Ultrafilter characterization of closure of an image ($b\in \overline{f[A]}⟺\exists \mathcal{H}(A\in \mathcal{H},f_{\ast }\mathcal{H}\to b)$)

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.