• Preservation of Limits. We have an equality of sets
    \[ f_{!}\webleft(\bigcap _{i\in I}U_{i}\webright)=\bigcap _{i\in I}f_{!}\webleft (U_{i}\webright ), \]

    natural in $\webleft\{ U_{i}\webright\} _{i\in I}\in \mathcal{P}\webleft (X\webright )^{\times I}$. In particular, we have equalities

    \[ \begin{gathered} f^{-1}\webleft (U\cap V\webright ) = f_{!}\webleft (U\webright )\cap f^{-1}\webleft (V\webright ),\\ f_{!}\webleft (X\webright ) = Y, \end{gathered} \]

    natural in $U,V\in \mathcal{P}\webleft (X\webright )$.


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