Item 1 (01FP) — The Clowder Project

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`

Functoriality. The assignment $\mathcal{U}\mapsto \bigcup _{U\in \mathcal{U}}U$ defines a functor
\[ \bigcup \colon \webleft (\mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright ),\subset \webright )\to \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ). \]

In particular, for each $\mathcal{U},\mathcal{V}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$, the following condition is satisfied:
- If $\mathcal{U}\subset \mathcal{V}$, then $\displaystyle \bigcup _{U\in \mathcal{U}}U\subset \bigcup _{V\in \mathcal{V}}V$.

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