This is a follow-up question to my previous question, [_Existence of a \*really\* nice topology on the powerset of a topological space_](https://mathoverflow.net/questions/441087/existence-of-a-really-nice-topology-on-the-powerset-of-a-topological-space), which, in a few words, asked about whether given a topological space $X$ we could induce a _compatible_ topology on its powerset $\mathcal{P}(X)$ in such a way that all the usual operations from set theory (unions, intersections, differences, etc.) become continuous with respect to this topology.

In the hope to make this question more self-contained, I'll repeat the relevant conditions stated there:
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.
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*}

There, James Hanson [proved](https://mathoverflow.net/a/441689) the following inexistence result, which I paraphrase:
> **Theorem.** Let $X$ be a topological space. If $(\mathcal{P}(X),\tau_1)$ and $(\mathcal{P}(\mathcal{P}(X)),\tau_2)$ satisfy conditions 1–6 and $\tau_1$ is not indiscrete, then $X$ is discrete.

Now, looking at the case of the Vietoris topology (arguably the most commonly used powerset topology), I noticed that condition 7 is misguided: when $\mathcal{P}(X)$ is equipped with the Vietoris topology, the inverse image function
$$f^{-1}\colon\mathcal{P}(Y)\to\mathcal{P}(X)$$
is continuous as a relation iff $f$ is closed and open, and the continuity of $f$ is in fact irrelevant. Meanwhile, the continuity of $f$ still implies that the direct image $f_*$ of $f$ is continuous, but again only as a relation. (See [the references here](https://mathoverflow.net/a/440811) for an explanation of the not-so-well-known notion of continuity of relations).

Thus, instead of condition 6 above, a more reasonable set of conditions would be to ask that just the direct image $f_*$ is continuous (either as a relation or ideally as a function).

------

**Question I.** Is there an interesting topology on $\mathcal{P}(X)$ satisfying conditions 2–4? What about 1–4?

**Question II.** Moreover, is there _a_ finest such topology in the sense of being a maximal element in the sublattice of all topologies on $\mathcal{P}(X)$, ordered by inclusion, spanned by those satisfying 2–4 (resp. 1–4)? Must such a topology be in fact a maximum element of this sublattice?

P.S. The minimal such element is the indiscrete topology.

**Question III.** Is there a general procedure of inducing interesting topologies on powersets (like the definition of the Vietoris topology) in such a way that 1–5 hold, and also $f_*$ is continuous¹ whenever $f$ is so?

(¹Either as a relation or as a function, I'm interested in both questions)

Lastly, note that a constraint on any topology satisfying 2–4 is that such a topology must make $\mathcal{P}(X)$ into a homogeneous topological space, as again  [proved](https://mathoverflow.net/a/441689) by James Hanson.