2.3.6 Unions of Families of Subsets

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

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

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

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:

    • If $\mathcal{U}\subset \mathcal{V}$, then $\displaystyle \bigcup _{U\in \mathcal{U}}U\subset \bigcup _{V\in \mathcal{V}}V$.

  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

    \[ \bigcup _{V\in \webleft\{ U\webright\} }V=U \]

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

  4. Right Unitality. The diagram

    commutes, i.e. we have

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

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

  5. Interaction With Unions I. The diagram

    commutes, i.e. we have

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

    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 \webleft(\bigcup _{V\in \mathcal{V}}V\webright) & = \bigcup _{V\in \mathcal{V}}\webleft (U\cup V\webright ),\\ \webleft(\bigcup _{U\in \mathcal{U}}U\webright)\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

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

    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 \webleft(\bigcup _{V\in \mathcal{V}}V\webright) & = \bigcup _{V\in \mathcal{V}}\webleft (U\cup V\webright ),\\ \webleft(\bigcup _{U\in \mathcal{U}}U\webright)\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 $U,V\in \mathcal{P}\webleft (X\webright )$.

  9. Interaction With Differences. The diagram

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

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

    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

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

    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

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

    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

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

    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

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

    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

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

    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

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

    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 Direct Images With Compact Support. 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

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

    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

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

    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.

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}}}=}}\webleft\{ 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} \webright\} \\ & = \webleft\{ 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} \webright\} \\ & = \webleft\{ 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} \webright\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\bigcup _{A\in \mathcal{A}}\webleft(\bigcup _{U\in A}U\webright). \end{align*}

This finishes the proof.

Item 3: Left Unitality
We have
\begin{align*} \bigcup _{V\in \webleft\{ U\webright\} }V & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ x\in X\ \middle |\ \begin{aligned} & \text{there exists some $V\in \webleft\{ U\webright\} $}\\ & \text{such that we have $x\in U$} \end{aligned} \webright\} \\ & = \webleft\{ x\in X\ \middle |\ x\in U \webright\} \\ & = U.\end{align*}

This finishes the proof.

Item 4: Right Unitality
We have
\begin{align*} \bigcup _{\webleft\{ u\webright\} \in \chi _{X}\webleft (U\webright )}\webleft\{ u\webright\} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ x\in X\ \middle |\ \begin{aligned} & \text{there exists some $\webleft\{ u\webright\} \in \chi _{X}\webleft (U\webright )$}\\ & \text{such that we have $x\in \webleft\{ u\webright\} $} \end{aligned} \webright\} \\ & = \webleft\{ x\in X\ \middle |\ \begin{aligned} & \text{there exists some $\webleft\{ u\webright\} \in \chi _{X}\webleft (U\webright )$}\\ & \text{such that we have $x=u$} \end{aligned} \webright\} \\ & = \webleft\{ x\in X\ \middle |\ \begin{aligned} & \text{there exists some $u\in U$}\\ & \text{such that we have $x=u$} \end{aligned} \webright\} \\ & = \webleft\{ x\in X\ \middle |\ x\in U\webright\} \\ & = 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}}}=}}\webleft\{ 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} \webright\} \\ & = \webleft\{ 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} \webright\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ x\in X\ \middle |\ \begin{aligned} & \text{there exists some $W\in \mathcal{U}$}\\ & \text{such that we have $x\in W$} \end{aligned} \webright\} \\ & \phantom{={}}\mkern 4mu\cup \webleft\{ x\in X\ \middle |\ \begin{aligned} & \text{there exists some $W\in \mathcal{V}$}\\ & \text{such that we have $x\in W$} \end{aligned} \webright\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft(\bigcup _{W\in \mathcal{U}}W\webright)\cup \webleft(\bigcup _{W\in \mathcal{V}}W\webright)\\ & = \webleft(\bigcup _{U\in \mathcal{U}}U\webright)\cup \webleft(\bigcup _{V\in \mathcal{V}}V\webright). \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}}}=}}\webleft\{ 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} \webright\} \\ & \subset \webleft\{ 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} \webright\} \\ & = \webleft\{ x\in X\ \middle |\ \begin{aligned} & \text{there exists some $U\in \mathcal{U}$}\\ & \text{such that we have $x\in U$} \end{aligned} \webright\} \\ & \phantom{={}}\mkern 4mu\cup \webleft\{ x\in X\ \middle |\ \begin{aligned} & \text{there exists some $V\in \mathcal{V}$}\\ & \text{such that we have $x\in V$} \end{aligned} \webright\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft(\bigcup _{U\in \mathcal{U}}U\webright)\cap \webleft(\bigcup _{V\in \mathcal{V}}V\webright).\end{align*}

This finishes the proof.

Item 8: Interaction With Intersections II
Omitted.
Item 9: Interaction With Differences
Let $X=\webleft\{ 0,1\webright\} $, let $\mathcal{U}=\webleft\{ \webleft\{ 0,1\webright\} \webright\} $, and let $\mathcal{V}=\webleft\{ \webleft\{ 0\webright\} \webright\} $. We have

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

whereas

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

Thus we have

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

This finishes the proof.

Item 10: Interaction With Complements I
Let $X=\webleft\{ 0,1\webright\} $ and let $\mathcal{U}=\webleft\{ 0\webright\} $. We have
\begin{align*} \bigcup _{U\in \mathcal{U}^{\textsf{c}}}U & = \bigcup _{U\in \webleft\{ \text{Ø},\webleft\{ 1\webright\} ,\webleft\{ 0,1\webright\} \webright\} }U\\ & = \webleft\{ 0,1\webright\} , \end{align*}

whereas

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

Thus we have

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

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=\webleft\{ 0,1\webright\} $, let $\mathcal{U}=\webleft\{ \webleft\{ 0,1\webright\} \webright\} $, and let $\mathcal{V}=\webleft\{ \webleft\{ 0\webright\} ,\webleft\{ 0,1\webright\} \webright\} $. We have
\begin{align*} \bigcup _{W\in \mathcal{U}\mathbin {\triangle }\mathcal{V}}W & = \bigcup _{W\in \webleft\{ \webleft\{ 0\webright\} \webright\} }W\\ & = \webleft\{ 0\webright\} , \end{align*}

whereas

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

Thus we have

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

This finishes the proof.

Item 14: Interaction With Internal Homs I
This is a repetition of Item 7 of Proposition 2.4.7.1.3 and is proved there.
Item 15: Interaction With Internal Homs II
This is a repetition of Item 8 of Proposition 2.4.7.1.3 and is proved there.
Item 16: Interaction With Internal Homs III
This is a repetition of Item 9 of Proposition 2.4.7.1.3 and is proved there.
Item 17: Interaction With Direct Images
This is a repetition of Item 3 of Proposition 2.6.1.1.4 and is proved there.
Item 18: Interaction With Inverse Images
This is a repetition of Item 3 of Proposition 2.6.2.1.3 and is proved there.
Item 19: Interaction With Direct Images With Compact Support
This is a repetition of Item 3 of Proposition 2.6.3.1.6 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}}}=}}\webleft\{ 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} \webright\} \\ & = \webleft\{ 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} \webright\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\bigcap _{A\in \mathcal{A}}\webleft(\bigcap _{U\in A}U\webright).\end{align*}

This finishes the proof.

Item 21: Interaction With Intersections of Families II
Omitted.


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