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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 6 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 6 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 4 hours ago

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