Item 8 (01FW) — The Clowder Project

4.3.6 Unions of Families of Subsets

Let $X$ be a set and let $\mathcal{U}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.

Definition 4.3.6.1.1 ▶ Unions of Families of Subsets

The union of $\mathcal{U}$ is the set $\bigcup _{U\in \mathcal{U}}U$ defined by

\[ \bigcup _{U\in \mathcal{U}}U\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ x\in X\ \middle |\ \begin{aligned} & \text{there exists some $U\in \mathcal{U}$}\\ & \text{such that we have $x\in U$}\end{aligned} \right\} . \]

Proposition 4.3.6.1.2 ▶ Properties of Unions of Families of Subsets

Let $X$ be a set.

1.
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:
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2.
Associativity. The diagram
commutes, i.e. we have

\[ \bigcup _{U\in {\scriptsize \displaystyle \bigcup _{A\in \mathcal{A}}A}}U=\bigcup _{A\in \mathcal{A}}\webleft (\bigcup _{U\in A}U\webright ) \]

for each $\mathcal{A}\in \mathcal{P}\webleft (\mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )\webright )$.
3.
Left Unitality. The diagram
commutes, i.e. we have

\[ \bigcup _{V\in \left\{ U\right\} }V=U \]

for each $U\in \mathcal{P}\webleft (X\webright )$.
4.
Right Unitality. The diagram
commutes, i.e. we have

\[ \bigcup _{\left\{ u\right\} \in \chi _{X}\webleft (U\webright )}\left\{ u\right\} =U \]

for each $U\in \mathcal{P}\webleft (X\webright )$.
5.
Interaction With Unions I. The diagram
commutes, i.e. we have

\[ \bigcup _{W\in \mathcal{U}\cup \mathcal{V}}W=\left(\bigcup _{U\in \mathcal{U}}U\right)\cup \left(\bigcup _{V\in \mathcal{V}}V\right) \]

for each $\mathcal{U},\mathcal{V}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.
6.
Interaction With Unions II. The diagrams
commute, i.e. we have

\begin{align*} U\cup \left(\bigcup _{V\in \mathcal{V}}V\right) & = \bigcup _{V\in \mathcal{V}}\webleft (U\cup V\webright ),\\ \left(\bigcup _{U\in \mathcal{U}}U\right)\cup V & = \bigcup _{U\in \mathcal{U}}\webleft (U\cup V\webright )\end{align*}

for each $\mathcal{U},\mathcal{V}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$ and each $U,V\in \mathcal{P}\webleft (X\webright )$.
7.
Interaction With Intersections I. We have a natural transformation
with components

\[ \bigcup _{W\in \mathcal{U}\cap \mathcal{V}}W\subset \left(\bigcup _{U\in \mathcal{U}}U\right)\cap \left(\bigcup _{V\in \mathcal{V}}V\right) \]

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

8.
Interaction With Intersections II. The diagrams
commute, i.e. we have

\begin{align*} U\cup \left(\bigcup _{V\in \mathcal{V}}V\right) & = \bigcup _{V\in \mathcal{V}}\webleft (U\cup V\webright ),\\ \left(\bigcup _{U\in \mathcal{U}}U\right)\cup V & = \bigcup _{U\in \mathcal{U}}\webleft (U\cup V\webright )\end{align*}

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

Interaction With Differences. The diagram

does not commute in general, i.e. we may have

\[ \bigcup _{W\in \mathcal{U}\setminus \mathcal{V}}W\neq \left(\bigcup _{U\in \mathcal{U}}U\right)\setminus \left(\bigcup _{V\in \mathcal{V}}V\right) \]

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

10.

Interaction With Complements I. The diagram

does not commute in general, i.e. we may have

\[ \bigcup _{U\in \mathcal{U}^{\textsf{c}}}U\neq \bigcup _{U\in \mathcal{U}}U^{\textsf{c}} \]

in general, where $\mathcal{U}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.

11.

Interaction With Complements II. The diagram

commutes, i.e. we have

\[ \left(\bigcup _{U\in \mathcal{U}}U\right)^{\textsf{c}}=\bigcap _{U\in \mathcal{U}}U^{\textsf{c}} \]

for each $\mathcal{U}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.

12.

Interaction With Complements III. The diagram

commutes, i.e. we have

\[ \left(\bigcap _{U\in \mathcal{U}}U\right)^{\textsf{c}}=\bigcup _{U\in \mathcal{U}}U^{\textsf{c}} \]

for each $\mathcal{U}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.

13.

Interaction With Symmetric Differences. The diagram

does not commute in general, i.e. we may have

\[ \bigcup _{W\in \mathcal{U}\mathbin {\triangle }\mathcal{V}}W\neq \left(\bigcup _{U\in \mathcal{U}}U\right)\mathbin {\triangle }\left(\bigcup _{V\in \mathcal{V}}V\right) \]

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

14.

Interaction With Internal Homs I. The diagram

does not commute in general, i.e. we may have

\[ \bigcup _{W\in \webleft [\mathcal{U},\mathcal{V}\webright ]_{\mathcal{P}\webleft (X\webright )}}W\neq \left[\bigcup _{U\in \mathcal{U}}U,\bigcup _{V\in \mathcal{V}}V\right]_{X} \]

in general, where $\mathcal{U}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.

15.

Interaction With Internal Homs II. The diagram

commutes, i.e. we have

\[ \left[\bigcup _{U\in \mathcal{U}}U,V\right]_{X}= \bigcap _{U\in \mathcal{U}}\webleft [U,V\webright ]_{X} \]

for each $\mathcal{U}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$ and each $V\in \mathcal{P}\webleft (X\webright )$.

16.

Interaction With Internal Homs III. The diagram

commutes, i.e. we have

\[ \left[U,\bigcup _{V\in \mathcal{V}}V\right]_{X}= \bigcup _{V\in \mathcal{V}}\webleft [U,V\webright ]_{X} \]

for each $U\in \mathcal{P}\webleft (X\webright )$ and each $\mathcal{V}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.

17.

Interaction With Direct Images. Let $f\colon X\to Y$ be a map of sets. The diagram

commutes, i.e. we have

\[ \bigcup _{U\in \mathcal{U}}f_{!}\webleft (U\webright )=\bigcup _{V\in f_{!}\webleft (\mathcal{U}\webright )}V \]

for each $\mathcal{U}\in \mathcal{P}\webleft (X\webright )$, where $f_{!}\webleft (\mathcal{U}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (f_{!}\webright )_{!}\webleft (\mathcal{U}\webright )$.

18.

Interaction With Inverse Images. Let $f\colon X\to Y$ be a map of sets. The diagram

commutes, i.e. we have

\[ \bigcup _{V\in \mathcal{V}}f^{-1}\webleft (V\webright )=\bigcup _{U\in f^{-1}\webleft (\mathcal{U}\webright )}U \]

for each $\mathcal{V}\in \mathcal{P}\webleft (Y\webright )$, where $f^{-1}\webleft (\mathcal{V}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (f^{-1}\webright )^{-1}\webleft (\mathcal{V}\webright )$.

19.

Interaction With Codirect Images. Let $f\colon X\to Y$ be a map of sets. The diagram

commutes, i.e. we have

\[ \bigcup _{U\in \mathcal{U}}f_{*}\webleft (U\webright )=\bigcup _{V\in f_{*}\webleft (\mathcal{U}\webright )}V \]

for each $\mathcal{U}\in \mathcal{P}\webleft (X\webright )$, where $f_{*}\webleft (\mathcal{U}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (f_{*}\webright )_{*}\webleft (\mathcal{U}\webright )$.

20.

Interaction With Intersections of Families I. The diagram

commutes, i.e. we have

\[ \bigcap _{U\in {\scriptsize \displaystyle \bigcup _{A\in \mathcal{A}}A}}U=\bigcap _{A\in \mathcal{A}}\left(\bigcap _{U\in A}U\right) \]

for each $\mathcal{A}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.

21.

Interaction With Intersections of Families II. Let $X$ be a set and consider the compositions

given by

\[ \begin{aligned} \mathcal{A} & \mapsto \bigcup _{U\in {\scriptsize \displaystyle \bigcap _{A\in \mathcal{A}}}A}U,\\ \mathcal{A} & \mapsto \bigcup _{A\in \mathcal{A}}\left(\bigcap _{U\in A}U\right), \end{aligned} \quad \begin{aligned} \mathcal{A} & \mapsto \bigcap _{U\in {\scriptsize \displaystyle \bigcup _{A\in \mathcal{A}}A}}U,\\ \mathcal{A} & \mapsto \bigcap _{A\in \mathcal{A}}\left(\bigcup _{U\in A}U\right) \end{aligned} \]

for each $\mathcal{A}\in \mathcal{P}\webleft (\mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )\webright )$. We have the following inclusions:

All other possible inclusions fail to hold in general.

Proof of Proposition 4.3.6.1.2.

Item 1: Functoriality

Since $\mathcal{P}\webleft (X\webright )$ is posetal, it suffices to prove the condition $\webleft (\star \webright )$. So let $\mathcal{U},\mathcal{V}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$ with $\mathcal{U}\subset \mathcal{V}$. We claim that

\[ \bigcup _{U\in \mathcal{U}}U\subset \bigcup _{V\in \mathcal{V}}V. \]

Indeed, given $x\in \bigcup _{U\in \mathcal{U}}U$, there exists some $U\in \mathcal{U}$ such that $x\in U$, but since $\mathcal{U}\subset \mathcal{V}$, we have $U\in \mathcal{V}$ as well, and thus $x\in \bigcup _{V\in \mathcal{V}}V$, which gives our desired inclusion.

Item 2: Associativity

We have

\begin{align*} \bigcup _{U\in {\scriptsize \displaystyle \bigcup _{A\in \mathcal{A}}A}}U & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ x\in X\ \middle |\ \begin{aligned} & \text{there exists some $U\in \displaystyle \bigcup _{A\in \mathcal{A}}A$}\\ & \text{such that we have $x\in U$} \end{aligned} \right\} \\ & = \left\{ x\in X\ \middle |\ \begin{aligned} & \text{there exists some $A\in \mathcal{A}$}\\ & \text{and some $U\in A$ such that}\\ & \text{we have $x\in U$} \end{aligned} \right\} \\ & = \left\{ x\in X\ \middle |\ \begin{aligned} & \text{there exists some $A\in \mathcal{A}$}\\ & \text{such that we have $x\in \bigcup _{U\in A}U$} \end{aligned} \right\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\bigcup _{A\in \mathcal{A}}\left(\bigcup _{U\in A}U\right). \end{align*}

This finishes the proof.

Item 3: Left Unitality

We have

\begin{align*} \bigcup _{V\in \left\{ U\right\} }V & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ x\in X\ \middle |\ \begin{aligned} & \text{there exists some $V\in \left\{ U\right\} $}\\ & \text{such that we have $x\in U$} \end{aligned} \right\} \\ & = \left\{ x\in X\ \middle |\ x\in U \right\} \\ & = U.\end{align*}

This finishes the proof.

Item 4: Right Unitality

We have

\begin{align*} \bigcup _{\left\{ u\right\} \in \chi _{X}\webleft (U\webright )}\left\{ u\right\} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ x\in X\ \middle |\ \begin{aligned} & \text{there exists some $\left\{ u\right\} \in \chi _{X}\webleft (U\webright )$}\\ & \text{such that we have $x\in \left\{ u\right\} $} \end{aligned} \right\} \\ & = \left\{ x\in X\ \middle |\ \begin{aligned} & \text{there exists some $\left\{ u\right\} \in \chi _{X}\webleft (U\webright )$}\\ & \text{such that we have $x=u$} \end{aligned} \right\} \\ & = \left\{ x\in X\ \middle |\ \begin{aligned} & \text{there exists some $u\in U$}\\ & \text{such that we have $x=u$} \end{aligned} \right\} \\ & = \left\{ x\in X\ \middle |\ x\in U\right\} \\ & = U.\end{align*}

This finishes the proof.

Item 5: Interaction With Unions I

We have

\begin{align*} \bigcup _{W\in \mathcal{U}\cup \mathcal{V}}W & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ x\in X\ \middle |\ \begin{aligned} & \text{there exists some $W\in \mathcal{U}\cup \mathcal{V}$}\\ & \text{such that we have $x\in W$} \end{aligned} \right\} \\ & = \left\{ x\in X\ \middle |\ \begin{aligned} & \text{there exists some $W\in \mathcal{U}$ or some}\\ & \text{$W\in \mathcal{V}$ such that we have $x\in W$} \end{aligned} \right\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ x\in X\ \middle |\ \begin{aligned} & \text{there exists some $W\in \mathcal{U}$}\\ & \text{such that we have $x\in W$} \end{aligned} \right\} \\ & \phantom{={}}\mkern 4mu\cup \left\{ x\in X\ \middle |\ \begin{aligned} & \text{there exists some $W\in \mathcal{V}$}\\ & \text{such that we have $x\in W$} \end{aligned} \right\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left(\bigcup _{W\in \mathcal{U}}W\right)\cup \left(\bigcup _{W\in \mathcal{V}}W\right)\\ & = \left(\bigcup _{U\in \mathcal{U}}U\right)\cup \left(\bigcup _{V\in \mathcal{V}}V\right). \end{align*}

This finishes the proof.

Item 6: Interaction With Unions II

Omitted.

Item 7: Interaction With Intersections I

We have

\begin{align*} \bigcup _{W\in \mathcal{U}\cap \mathcal{V}}W & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ x\in X\ \middle |\ \begin{aligned} & \text{there exists some $W\in \mathcal{U}\cap \mathcal{V}$}\\ & \text{such that we have $x\in W$} \end{aligned} \right\} \\ & \subset \left\{ x\in X\ \middle |\ \begin{aligned} & \text{there exists some $U\in \mathcal{U}$ and some $V\in \mathcal{V}$}\\ & \text{such that we have $x\in U$ and $x\in V$} \end{aligned} \right\} \\ & = \left\{ x\in X\ \middle |\ \begin{aligned} & \text{there exists some $U\in \mathcal{U}$}\\ & \text{such that we have $x\in U$} \end{aligned} \right\} \\ & \phantom{={}}\mkern 4mu\cup \left\{ x\in X\ \middle |\ \begin{aligned} & \text{there exists some $V\in \mathcal{V}$}\\ & \text{such that we have $x\in V$} \end{aligned} \right\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left(\bigcup _{U\in \mathcal{U}}U\right)\cap \left(\bigcup _{V\in \mathcal{V}}V\right).\end{align*}

This finishes the proof.

Item 8: Interaction With Intersections II

Omitted.

Item 9: Interaction With Differences

Let $X=\left\{ 0,1\right\} $, let $\mathcal{U}=\left\{ \left\{ 0,1\right\} \right\} $, and let $\mathcal{V}=\left\{ \left\{ 0\right\} \right\} $. We have

\begin{align*} \bigcup _{W\in \mathcal{U}\setminus \mathcal{V}}U & = \bigcup _{W\in \left\{ \left\{ 0,1\right\} \right\} }W\\ & = \left\{ 0,1\right\} , \end{align*}

whereas

\begin{align*} \left(\bigcup _{U\in \mathcal{U}}U\right)\setminus \left(\bigcup _{V\in \mathcal{V}}V\right) & = \left\{ 0,1\right\} \setminus \left\{ 0\right\} \\ & = \left\{ 1\right\} . \end{align*}

Thus we have

\[ \bigcup _{W\in \mathcal{U}\setminus \mathcal{V}}W=\left\{ 0,1\right\} \neq \left\{ 1\right\} =\left(\bigcup _{U\in \mathcal{U}}U\right)\setminus \left(\bigcup _{V\in \mathcal{V}}V\right). \]

This finishes the proof.

Item 10: Interaction With Complements I

Let $X=\left\{ 0,1\right\} $ and let $\mathcal{U}=\left\{ 0\right\} $. We have

\begin{align*} \bigcup _{U\in \mathcal{U}^{\textsf{c}}}U & = \bigcup _{U\in \left\{ \text{Ø},\left\{ 1\right\} ,\left\{ 0,1\right\} \right\} }U\\ & = \left\{ 0,1\right\} , \end{align*}

whereas

\begin{align*} \bigcup _{U\in \mathcal{U}}U^{\textsf{c}} & = \left\{ 0\right\} ^{\textsf{c}}\\ & = \left\{ 1\right\} . \end{align*}

Thus we have

\[ \bigcup _{U\in \mathcal{U}^{\textsf{c}}}U=\left\{ 0,1\right\} \neq \left\{ 1\right\} =\bigcup _{U\in \mathcal{U}}U^{\textsf{c}}. \]

This finishes the proof.

Item 11: Interaction With Complements II

Omitted.

Item 12: Interaction With Complements III

Omitted.

Item 13: Interaction With Symmetric Differences

Let $X=\left\{ 0,1\right\} $, let $\mathcal{U}=\left\{ \left\{ 0,1\right\} \right\} $, and let $\mathcal{V}=\left\{ \left\{ 0\right\} ,\left\{ 0,1\right\} \right\} $. We have

\begin{align*} \bigcup _{W\in \mathcal{U}\mathbin {\triangle }\mathcal{V}}W & = \bigcup _{W\in \left\{ \left\{ 0\right\} \right\} }W\\ & = \left\{ 0\right\} , \end{align*}

whereas

\begin{align*} \left(\bigcup _{U\in \mathcal{U}}U\right)\mathbin {\triangle }\left(\bigcup _{V\in \mathcal{V}}V\right) & = \left\{ 0,1\right\} \mathbin {\triangle }\left\{ 0,1\right\} \\ & = \text{Ø}, \end{align*}

Thus we have

\[ \bigcup _{W\in \mathcal{U}\mathbin {\triangle }\mathcal{V}}W=\left\{ 0\right\} \neq \text{Ø}=\left(\bigcup _{U\in \mathcal{U}}U\right)\mathbin {\triangle }\left(\bigcup _{V\in \mathcal{V}}V\right). \]

This finishes the proof.

Item 14: Interaction With Internal Homs I

This is a repetition of Item 7 of Proposition 4.4.7.1.3 and is proved there.

Item 15: Interaction With Internal Homs II

This is a repetition of Item 8 of Proposition 4.4.7.1.3 and is proved there.

Item 16: Interaction With Internal Homs III

This is a repetition of Item 9 of Proposition 4.4.7.1.3 and is proved there.

Item 17: Interaction With Direct Images

This is a repetition of Item 3 of Proposition 4.6.1.1.5 and is proved there.

Item 18: Interaction With Inverse Images

This is a repetition of Item 3 of Proposition 4.6.2.1.3 and is proved there.

Item 19: Interaction With Codirect Images

This is a repetition of Item 3 of Proposition 4.6.3.1.7 and is proved there.

Item 20: Interaction With Intersections of Families I

We have

\begin{align*} \bigcap _{U\in {\scriptsize \displaystyle \bigcup _{A\in \mathcal{A}}A}}U & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ x\in X\ \middle |\ \begin{aligned} & \text{for each $U\in \bigcup _{A\in \mathcal{A}}A$,}\\ & \text{we have $x\in U$} \end{aligned} \right\} \\ & = \left\{ x\in X\ \middle |\ \begin{aligned} & \text{for each $A\in \mathcal{A}$ and each}\\ & \text{$U\in A$, we have $x\in U$} \end{aligned} \right\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\bigcap _{A\in \mathcal{A}}\left(\bigcap _{U\in A}U\right).\end{align*}

This finishes the proof.

Item 21: Interaction With Intersections of Families II

Omitted.

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