= Pushforward filter
{title2=$f_*\mathcal F=\{B:f^{-1}(B)\in\mathcal F\}$}
= Image of a filter
{synonym}
= Images of a filter
{synonym}
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>.
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