= Ultrafilter characterization of closure of an image
{title2=$b\in\overline{f[A]}\iff\exists\mathcal H\ (A\in\mathcal H,\ f_*\mathcal H\to b)$}
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.
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