Ways to induce a topology on power set?

You can mimic the construction of various hyperspaces to extend beyond the closed subsets of a space.

To introduce some notation, given a set $X$ and a subset $A \subseteq X$, we define $$ [A]^+ = \{ Z \subseteq X : Z \subseteq A \}; \\ [A]^- = \{ Z \subseteq X : Z \cap A \neq \varnothing \} . $$

For a couple of examples, let's fix a topological space $X$:

1. Consider the topology on $\mathcal{P}(X)$ generated by sets of the form

   • $[ U ]^+$ for open $U \subseteq X$; and
   • $[ U ]^-$ for open $U \subseteq X$.

   The subspace of this space consisting of the closed subsets of $X$ is called the Vietoris (or finite or exponential) topology.

2. Consider the topology on $\mathcal{P} (X)$ generated by sets of the form

   • $[X \setminus K]^+$ for compact $K \subseteq X$; and
   • $[U]^-$ for open $U \subseteq X$.

   The subspace of this space consisting of the closed subsets of $X$ is called the Fell topology. (Of particular note, the Fell topology is always compact (though not necessarily Hausdorff).)

Let a set be open iff it is empty or of the form $\mathcal{P}(X)\backslash\big\{\{x\}:x\in C\big\}$ for some closed set C.