Item 4 (008V) — The Clowder Project

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Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

*To be replaced with Linus Romer's Elemaints when it is released.

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

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

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

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