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Footnotes.

¹The motivation for this is that $\circledast$ is the zero-categorical analogue of ordinary Day convolution, and we may compute it via a completely analogous coend formula when viewing subsets $U$ of $X$ as functions $\chi_U\colon X\to\{\mathrm{true},\mathrm{false}\}$, namely $\chi_U\circledast\chi_V=\int^{x,y\in X}\mathrm{Hom}_{X}(-,xy)\times\chi_U(x)\times\chi_V(y)$.

TL;DR. Given a topological space $X$, is there a natural way to "induce" a topology on $\mathcal{P}(X)$ from the topology of $X$ in such a way that 1) all the basic operations of set theory (intersections, unions, direct images, etc.) become continuous and 2) the topologies on $X$ and $\mathcal{P}(X)$ are compatible in a certain sense?

Given a topological space $X$, there are a couple of ways to assign a topology to its powerset $\mathcal{P}(X)$ that do not depend specifically on $X$. For example:

• We can view $\mathcal{P}(X)$ as the hom $\mathrm{Hom}(X,2)$, put either the discrete, indiscrete, or Sierpiński topology on $2$, and consider the compact-open topology;
• We can take the Alexandroff topology with respect to $\subset$ or $\supset$;
• There are the so-called hit-and-miss topologies, of which the Vietoris and Fell topologies are examples.
• (A non-example would be to define a topology on $\mathcal{P}(\mathbb{R})$ using the order of $\mathbb{R}$.)

Given an assignment of topologies on $\mathcal{P}(X)$ from topologies on $X$ as above, we could consider the following sets of niceness conditions:

1. The map $\iota\colon X\to\mathcal{P}(X)$ given by $x\mapsto\{x\}$ is continuous.
2. Binary unions $\cup\colon\mathcal{P}(X)\times\mathcal{P}(X)\to\mathcal{P}(X)$ are continuous
3. Binary intersections $\cap\colon\mathcal{P}(X)\times\mathcal{P}(X)\to\mathcal{P}(X)$ are continuous
4. Differences $\setminus\colon\mathcal{P}(X)\times\mathcal{P}(X)\to\mathcal{P}(X)$ are continuous.
5. Arbitrary unions $\bigcup\colon\mathcal{P}(\mathcal{P}(X))\to\mathcal{P}(X)$ are continuous.
6. Arbitrary intersections $\bigcap\colon\mathcal{P}(\mathcal{P}(X))\to\mathcal{P}(X)$ are continuous.
7. If $f\colon X\to Y$ is a continuous map of topological spaces, then so are its direct and inverse images \begin{align*} f_{*} &\colon\mathcal{P}(X)\to\mathcal{P}(Y),\\ f^{-1} &\colon\mathcal{P}(Y)\to\mathcal{P}(X). \end{align*}

I'm also interested in the following two sets of additional conditions that are a bit more specific:

More compatibility conditions between $X$ and $\mathcal{P}(X)$:

8. For each $x\in X$, if $\{x\}$ is closed in $X$, then $\{\{x\}\}$ is closed in $\mathcal{P}(X)$.
9. If $S\subset X$ is closed in $X$, then $\{S\}$ is closed in $\mathcal{P}(X)$.
10. If $U\subset X$ is open in $X$, then $\{U\}$ is open in $\mathcal{P}(X)$.

Another niceness requirement for the topology on $\mathcal{P}(X)$:

11. Given any monoid structure $(\star,1_X)$ on $X$ making it into a topological monoid, the map $$\circledast\colon\mathcal{P}(X)\times\mathcal{P}(X)\to\mathcal{P}(X)$$ given by $$U\circledast V:=\{uv\in X\ |\ u\in U,v\in V\}$$ is continuous.¹
12. (Implies 11 by Tobias's second comment). Given topological spaces $X$ and $Y$, the map $$\mathcal{P}(X)\times\mathcal{P}(Y)\to\mathcal{P}(X\times Y)$$ given by $(U,V)\mapsto U\times V$ is continuous.
13. Given topological spaces $X$ and $Y$, the bijection $$\mathcal{P}(X)\times\mathcal{P}(Y)\to\mathcal{P}(X\sqcup Y)$$ given by $(U,V)\mapsto U\cup V$ is a homeomorphism.
14. Given topological spaces $X$ and $Y$, the isomorphism of suplattices $$\mathcal{P}(X)\otimes\mathcal{P}(Y)\to\mathcal{P}(X\times Y)$$ is a homeomorphism, where $\otimes$ denotes the tensor product of suplattices.

Question I. Does there exist a powerset topology satisfying (at least but not necessarily only) conditions 1–7? What about (1–7+11), 1–8, 1–9, or 1–10?

Question II. Given a topological space $X$, what is the finest topology on $\mathcal{P}(X)$ satisfying conditions 1–6? What about (1–6 + 11–14), or these plus any of 8–10? Lastly, in case this topology turns out to be definable in a way that is independent of $X$ (like the Vietoris topology), do we also have 7?

TL;DR. Given a topological space $X$, is there a natural way to "induce" a topology on $\mathcal{P}(X)$ from the topology of $X$ in such a way that 1) all the basic operations of set theory (intersections, unions, direct images, etc.) become continuous and 2) the topologies on $X$ and $\mathcal{P}(X)$ are compatible in a certain sense?

Given a topological space $X$, there are a couple of ways to assign a topology to its powerset $\mathcal{P}(X)$ that do not depend specifically on $X$. For example:

• We can view $\mathcal{P}(X)$ as the hom $\operatorname{Hom}(X,2)$, put either the discrete, indiscrete, or Sierpiński topology on $2$, and consider the compact-open topology;
• We can take the Alexandroff topology with respect to $\subset$ or $\supset$;
• There are the so-called hit-and-miss topologies, of which the Vietoris and Fell topologies are examples.
• (A non-example would be to define a topology on $\mathcal{P}(\mathbb{R})$ using the order of $\mathbb{R}$.)

Given an assignment of topologies on $\mathcal{P}(X)$ from topologies on $X$ as above, we could consider the following sets of niceness conditions:

1. The map $\iota\colon X\to\mathcal{P}(X)$ given by $x\mapsto\{x\}$ is continuous.
2. Binary union ${\cup}\colon\mathcal{P}(X)\times\mathcal{P}(X)\to\mathcal{P}(X)$ is continuous.
3. Binary intersection ${\cap}\colon\mathcal{P}(X)\times\mathcal{P}(X)\to\mathcal{P}(X)$ is continuous.
4. Difference ${\setminus}\colon\mathcal{P}(X)\times\mathcal{P}(X)\to\mathcal{P}(X)$ is continuous.
5. Arbitrary union ${\bigcup}\colon\mathcal{P}(\mathcal{P}(X))\to\mathcal{P}(X)$ is continuous.
6. Arbitrary intersection ${\bigcap}\colon\mathcal{P}(\mathcal{P}(X))\to\mathcal{P}(X)$ is continuous.
7. If $f\colon X\to Y$ is a continuous map of topological spaces, then so are its direct and inverse images \begin{align*} f_{*} &{}\colon\mathcal{P}(X)\to\mathcal{P}(Y),\\ f^{-1} &{}\colon\mathcal{P}(Y)\to\mathcal{P}(X). \end{align*}

I'm also interested in the following two sets of additional conditions that are a bit more specific:

More compatibility conditions between $X$ and $\mathcal{P}(X)$:

8. For each $x\in X$, if $\{x\}$ is closed in $X$, then $\{\{x\}\}$ is closed in $\mathcal{P}(X)$.
9. If $S\subset X$ is closed in $X$, then $\{S\}$ is closed in $\mathcal{P}(X)$.
10. If $U\subset X$ is open in $X$, then $\{U\}$ is open in $\mathcal{P}(X)$.

Another niceness requirement for the topology on $\mathcal{P}(X)$:

11. Given any monoid structure $(\star,1_X)$ on $X$ making it into a topological monoid, the map $$\circledast\colon\mathcal{P}(X)\times\mathcal{P}(X)\to\mathcal{P}(X)$$ given by $$U\circledast V:=\{uv\in X\ |\ u\in U,v\in V\}$$ is continuous.¹
12. (Implies 11 by Tobias's second comment.) Given topological spaces $X$ and $Y$, the map $$\mathcal{P}(X)\times\mathcal{P}(Y)\to\mathcal{P}(X\times Y)$$ given by $(U,V)\mapsto U\times V$ is continuous.
13. Given topological spaces $X$ and $Y$, the bijection $$\mathcal{P}(X)\times\mathcal{P}(Y)\to\mathcal{P}(X\sqcup Y)$$ given by $(U,V)\mapsto U\cup V$ is a homeomorphism.
14. Given topological spaces $X$ and $Y$, the isomorphism of suplattices $$\mathcal{P}(X)\otimes\mathcal{P}(Y)\to\mathcal{P}(X\times Y)$$ is a homeomorphism, where $\otimes$ denotes the tensor product of suplattices (see Eric Wofsey's answer to Concrete description of the tensor product of suplattices?).

Question I. Does there exist a powerset topology satisfying (at least but not necessarily only) conditions 1–7? What about (1–7+11), 1–8, 1–9, or 1–10?

Question II. Given a topological space $X$, what is the finest topology on $\mathcal{P}(X)$ satisfying conditions 1–6? What about (1–6 + 11–14), or these plus any of 8–10? Lastly, in case this topology turns out to be definable in a way that is independent of $X$ (like the Vietoris topology), do we also have 7?

TL;DR. Given a topological space $X$, is there a natural way to "induce" a topology on $\mathcal{P}(X)$ from the topology of $X$ in such a way that 1) all the basic operations of set theory (intersections, unions, direct images, etc.) become continuous and 2) the topologies on $X$ and $\mathcal{P}(X)$ are compatible in a certain sense?

Given a topological space $X$, there are a couple of ways to assign a topology to its powerset $\mathcal{P}(X)$ that do not depend specifically on $X$. For example:

• We can view $\mathcal{P}(X)$ as the hom $\mathrm{Hom}(X,2)$, put either the discrete, indiscrete, or Sierpiński topology on $2$, and consider the compact-open topology;
• We can take the Alexandroff topology with respect to $\subset$ or $\supset$;
• There are the so-called hit-and-miss topologies, of which the Vietoris and Fell topologies are examples.
• (A non-example would be to define a topology on $\mathcal{P}(\mathbb{R})$ using the order of $\mathbb{R}$.)

Given an assignment of topologies on $\mathcal{P}(X)$ from topologies on $X$ as above, we could consider the following sets of niceness conditions:

1. The map $\iota\colon X\to\mathcal{P}(X)$ given by $x\mapsto\{x\}$ is continuous.
2. Binary unions $\cup\colon\mathcal{P}(X)\times\mathcal{P}(X)\to\mathcal{P}(X)$ are continuous
3. Binary intersections $\cap\colon\mathcal{P}(X)\times\mathcal{P}(X)\to\mathcal{P}(X)$ are continuous
4. Differences $\setminus\colon\mathcal{P}(X)\times\mathcal{P}(X)\to\mathcal{P}(X)$ are continuous.
5. Arbitrary unions $\bigcup\colon\mathcal{P}(\mathcal{P}(X))\to\mathcal{P}(X)$ are continuous.
6. Arbitrary intersections $\bigcap\colon\mathcal{P}(\mathcal{P}(X))\to\mathcal{P}(X)$ are continuous.
7. If $f\colon X\to Y$ is a continuous map of topological spaces, then so are its direct and inverse images \begin{align*} f_{*} &\colon\mathcal{P}(X)\to\mathcal{P}(Y),\\ f^{-1} &\colon\mathcal{P}(Y)\to\mathcal{P}(X). \end{align*}

I'm also interested in the following two sets of additional conditions that are a bit more specific:

More compatibility conditions between $X$ and $\mathcal{P}(X)$:

8. For each $x\in X$, if $\{x\}$ is closed in $X$, then $\{\{x\}\}$ is closed in $\mathcal{P}(X)$.
9. If $S\subset X$ is closed in $X$, then $\{S\}$ is closed in $\mathcal{P}(X)$.
10. If $U\subset X$ is open in $X$, then $\{U\}$ is open in $\mathcal{P}(X)$.

Another niceness requirement for the topology on $\mathcal{P}(X)$:

11. Given any monoid structure $(\star,1_X)$ on $X$ making it into a topological monoid, the map $$\circledast\colon\mathcal{P}(X)\times\mathcal{P}(X)\to\mathcal{P}(X)$$ given by $$U\circledast V:=\{uv\in X\ |\ u\in U,v\in V\}$$ is continuous.¹

Question. Does there exist a powerset topology satisfying (at least but not necessarily only) conditions 1–7? What about (1–7+11), 1–8, 1–9, or 1–10?

TL;DR. Given a topological space $X$, is there a natural way to "induce" a topology on $\mathcal{P}(X)$ from the topology of $X$ in such a way that 1) all the basic operations of set theory (intersections, unions, direct images, etc.) become continuous and 2) the topologies on $X$ and $\mathcal{P}(X)$ are compatible in a certain sense?

Given a topological space $X$, there are a couple of ways to assign a topology to its powerset $\mathcal{P}(X)$ that do not depend specifically on $X$. For example:

• We can view $\mathcal{P}(X)$ as the hom $\mathrm{Hom}(X,2)$, put either the discrete, indiscrete, or Sierpiński topology on $2$, and consider the compact-open topology;
• We can take the Alexandroff topology with respect to $\subset$ or $\supset$;
• There are the so-called hit-and-miss topologies, of which the Vietoris and Fell topologies are examples.
• (A non-example would be to define a topology on $\mathcal{P}(\mathbb{R})$ using the order of $\mathbb{R}$.)

Given an assignment of topologies on $\mathcal{P}(X)$ from topologies on $X$ as above, we could consider the following sets of niceness conditions:

1. The map $\iota\colon X\to\mathcal{P}(X)$ given by $x\mapsto\{x\}$ is continuous.
2. Binary unions $\cup\colon\mathcal{P}(X)\times\mathcal{P}(X)\to\mathcal{P}(X)$ are continuous
3. Binary intersections $\cap\colon\mathcal{P}(X)\times\mathcal{P}(X)\to\mathcal{P}(X)$ are continuous
4. Differences $\setminus\colon\mathcal{P}(X)\times\mathcal{P}(X)\to\mathcal{P}(X)$ are continuous.
5. Arbitrary unions $\bigcup\colon\mathcal{P}(\mathcal{P}(X))\to\mathcal{P}(X)$ are continuous.
6. Arbitrary intersections $\bigcap\colon\mathcal{P}(\mathcal{P}(X))\to\mathcal{P}(X)$ are continuous.
7. If $f\colon X\to Y$ is a continuous map of topological spaces, then so are its direct and inverse images \begin{align*} f_{*} &\colon\mathcal{P}(X)\to\mathcal{P}(Y),\\ f^{-1} &\colon\mathcal{P}(Y)\to\mathcal{P}(X). \end{align*}

I'm also interested in the following two sets of additional conditions that are a bit more specific:

More compatibility conditions between $X$ and $\mathcal{P}(X)$:

8. For each $x\in X$, if $\{x\}$ is closed in $X$, then $\{\{x\}\}$ is closed in $\mathcal{P}(X)$.
9. If $S\subset X$ is closed in $X$, then $\{S\}$ is closed in $\mathcal{P}(X)$.
10. If $U\subset X$ is open in $X$, then $\{U\}$ is open in $\mathcal{P}(X)$.

Another niceness requirement for the topology on $\mathcal{P}(X)$:

11. Given any monoid structure $(\star,1_X)$ on $X$ making it into a topological monoid, the map $$\circledast\colon\mathcal{P}(X)\times\mathcal{P}(X)\to\mathcal{P}(X)$$ given by $$U\circledast V:=\{uv\in X\ |\ u\in U,v\in V\}$$ is continuous.¹
12. (Implies 11 by Tobias's second comment). Given topological spaces $X$ and $Y$, the map $$\mathcal{P}(X)\times\mathcal{P}(Y)\to\mathcal{P}(X\times Y)$$ given by $(U,V)\mapsto U\times V$ is continuous.
13. Given topological spaces $X$ and $Y$, the bijection $$\mathcal{P}(X)\times\mathcal{P}(Y)\to\mathcal{P}(X\sqcup Y)$$ given by $(U,V)\mapsto U\cup V$ is a homeomorphism.
14. Given topological spaces $X$ and $Y$, the isomorphism of suplattices $$\mathcal{P}(X)\otimes\mathcal{P}(Y)\to\mathcal{P}(X\times Y)$$ is a homeomorphism, where $\otimes$ denotes the tensor product of suplattices.

Question I. Does there exist a powerset topology satisfying (at least but not necessarily only) conditions 1–7? What about (1–7+11), 1–8, 1–9, or 1–10?

Question II. Given a topological space $X$, what is the finest topology on $\mathcal{P}(X)$ satisfying conditions 1–6? What about (1–6 + 11–14), or these plus any of 8–10? Lastly, in case this topology turns out to be definable in a way that is independent of $X$ (like the Vietoris topology), do we also have 7?