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This question is an extension/clarification of the question: Is there a natural topology for sets of topological spaces?

The Hausdorff distance assigns a distance to any two subspaces $X, Y$ of a fixed metric space $M$. I am interested in the topological analog of this distance.

Question: is a natural topological structure on the set of subspaces of a fixed topological space $M$?

Note that there are no set-theoretic issues since the collection of subsets of a given set form a set.

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    $\begingroup$ en.m.wikipedia.org/wiki/Hypertopology $\endgroup$
    – Steven Clontz
    Commented 11 hours ago
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    $\begingroup$ You could identify every subset of $X$ with its characteristic function $X\to \mathbb 2$, and put on $P(X)$ the compact open topology of this set of mappings. $\endgroup$
    – Pietro Majer
    Commented 11 hours ago
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    $\begingroup$ @PietroMajer Indeed one could do that but since there aren't that many topologies on 2, this would basically be saying that an open subset of $P(X)$ consists of subsets of X that agree on various compacts. This seems quite restrictive and doesn't really generalize the Hausdorff distance. $\endgroup$
    – user39598
    Commented 9 hours ago
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    $\begingroup$ I'm currently writing a chapter on exactly this, but it will take a long while until everything is done. Here's a preliminary version. $\endgroup$
    – Emily
    Commented 4 hours ago
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    $\begingroup$ I also recommend this book and this paper, both of which talk about the Vietoris topology, one of the more common topologies on sets of subsets, on all of $\mathcal{P}(X)$, rather than only on the set $\mathrm{Cld}(X)$ of closed subspaces of $X$. $\endgroup$
    – Emily
    Commented 4 hours ago

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