Ways to induce a topology on power set? 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 proved there that neither such construction need be a topology on $\mathcal P(X)$ in general. (The latter will be such a topology if and only if $\mathcal T$ is discrete. If we know that $\mathcal T$ is $T_1$, then the former will be a topology if and only if $\mathcal T$ is discrete.)
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 do so is to give $\mathcal P(X)$ the topology of pointwise convergence of indicator functions $X\to\{0,1\}.$ This is certainly very nice, but I'm still curious:

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

(As a bonus question, can anyone can think of any non-$T_1$ topologies for which the first construction described above is a topology, or a proof that no such non-$T_1$ topology can exist? I will gladly upvote any such example/proof and link to it from my answer to the question above.)

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.
