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

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

    \[ \begin{gathered} f_{*}\webleft (U\webright )\cup f_{*}\webleft (V\webright ) = f_{*}\webleft (U\cup V\webright ),\\ f_{*}\webleft (\text{Ø}\webright ) = \text{Ø}, \end{gathered} \]

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


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