Ways to induce a topology on power set?

Question

In this question, two potential topologies were proposed for the power set of a set $X$ with a topology $\mathcal T$: one comprised of all sets of subsets of $X$ whose union was $\mathcal T$-open, one comprised of all sets of subsets of $X$ whose intersection was $\mathcal T$-open. I outlined a proof there that neither such construction need be a topology on $\mathcal P(X)$ in general. In fact, depending on how one interprets the intersection of the empty set, each will be a topology on $\mathcal P(X)$ if and only if $\mathcal T$ is the discrete topology on $X,$ in which case both are the discrete topology on $\mathcal P(X).$

This led me to wonder if there are any ways to induce a topology on $\mathcal P(X)$ from a topology on $X$? Some searching shows that one "natural" way to topologize $\mathcal P(X)$ is to give it the topology of pointwise convergence of indicator functions $X\to\{0,1\}.$ This is certainly very nice, but ignores the topology on $X,$ so I'm still curious:

Are there any ways to induce a topology on $\mathcal P(X)$ from any given topology on $X$? Ideally, I would like for different topologies on $X$ to give rise to different (though potentially homeomorphic, of course) topologies on $\mathcal P(X).$

Answer 1

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.

Answer 2

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).)