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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.)

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  • $\begingroup$ I have a feeling that this could get out of hand... ;) Fun question! $\endgroup$ – Bruno Joyal Feb 6 '14 at 2:46
  • $\begingroup$ @downvoter: Care to offer a critique? Perhaps request some improvement to my post or some clarification from me? $\endgroup$ – Cameron Buie Dec 15 '14 at 19:31
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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.

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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).)

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