The direct image function associated to $f$ is the function
\[ f_{*}\colon \mathcal{P}\webleft (X\webright )\to \mathcal{P}\webleft (Y\webright ) \]
defined by1
\begin{align*} f_{*}\webleft (U\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f\webleft (U\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ y\in Y\ \middle |\ \begin{aligned} & \text{there exists some $x\in U$}\\ & \text{such that $y=f\webleft (x\webright )$} \end{aligned} \webright\} \\ & = \webleft\{ f\webleft (x\webright )\in Y\ \middle |\ x\in U\webright\} \end{align*}
for each $U\in \mathcal{P}\webleft (X\webright )$.
1Further Terminology: The set $f\webleft (U\webright )$ is called the direct image of $U$ by $f$.
