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The Gromov-Hausdorff metric makes the set of compact metric spaces into a metric space itself. I am wondering what some natural generalizations there are for arbitrary topological spaces. Namely, is there a natural topology on the set of (compact?) topological spaces?

Edit: I am not too concerned about set-theoretic issues, but perhaps part of the problem is to find a special collection of topological spaces that do form a set and have a natural topological structure. I am more interested in the topological structure.

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    $\begingroup$ The compact topological spaces do not form a set but a proper class, so you can not expect traditional topological structures on it. There are alternatives however, like Grothendieck topology. $\endgroup$
    – Zerox
    Commented 19 hours ago
  • $\begingroup$ @Zerox you could add this as an answer. $\endgroup$
    – David Roberts ♦
    Commented 10 hours ago
  • $\begingroup$ The following appears in Continuity and Baire functions by Edgar Raymond Lorch [American Mathematical Monthly 78 #7 (Aug-Sep 1971), pp. 748-762]: "This circumstance, that a collection of topologies is topologized, may seem a bit incestuous." Slightly more detail in this mathoverflow answer. $\endgroup$
    – Dave L Renfro
    Commented 8 hours ago
  • $\begingroup$ I do not understand the question: Are you fixing a set $Y$ and looking for a topology on the set of topological spaces $(X,T)$, where $X\in 2^Y$? If so, what properties do you want this topology to have? Do you have a particular application in mind? $\endgroup$
    – Moishe Kohan
    Commented 2 hours ago
  • $\begingroup$ If you consider just the set of compact metric spaces and drop the isometry condition in the definition of the Gromov-Hausdorff metric all spaces have distance zero because you can make homeomorphic copies have arbitrarily small diameter in the Hilbert cube, and have these copies converge to the origin. You may want to try how you'd overcome this problem just in this particular case. $\endgroup$
    – KP Hart
    Commented 32 mins ago

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There is no set of all [compact] topological spaces.

Given a set of topological spaces, consider its power set. This power set with the indiscrete topology is a compact topological space missing from the set.

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  • $\begingroup$ If one came up with a decent notion of what it means for a sequence of topological spaces to converge to another topological space, then these set theoretic issues wouldn’t really be an issue; one could try to put a topology on any given set of spaces in which convergence behaves in the desired way. Maybe there would be some sequences (of spaces in the set) whose limit exists but is missing from the set, but that would be okay. On the other hand I don’t have a suggestion for such a notion of convergence… $\endgroup$
    – Dan Ramras
    Commented 11 hours ago
  • $\begingroup$ @DanRamras To get a full description of the topological structure it is not enough to only look at convergent sequences, but convergent nets, which does involve set-theoretic issues I think. $\endgroup$
    – Zerox
    Commented 3 hours ago

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