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Let $(X,\tau)$ be a topological space. Is there anything like "dual space" whose points are members of $\tau$? Would there be a natural way to define the topology on $\tau$?

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    $\begingroup$ Um,... you just did? $\endgroup$ – fleablood Nov 10 '17 at 18:34
  • $\begingroup$ This sounds like the beginnings of the construction of Čech homology/cohomology ... $\endgroup$ – Ted Shifrin Nov 10 '17 at 18:38
  • $\begingroup$ @TedShifrin do you mean Stone-Čech compactification? $\endgroup$ – ziggurism Nov 15 '17 at 15:41
  • $\begingroup$ @fleablood to define a topology you have to specify the open sets, not just the points. $\endgroup$ – Nathaniel Jan 21 at 5:45
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In Stone duality, given a Boolean algebra $A$, the homomorphisms $\text{hom}(A,2)$ are the ultrafilters and make up the Stone space of $A$. Ultrafilters on $A$ is not the same thing as open sets, but they're related. And the sets that are both closed and open are in correspondence with $A$ again.

There are other versions of this duality. The $C^*$ algebra of continuous functions on a compact Hausdorff space. The embedding of a space into its Stone-Čech compactification. Maximal ideals of the continuous functions on a variety with Zariski topology.

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  • $\begingroup$ I'm looking for something where the space with underlying set $\tau$ is, of course, related to the original space, but has potential to be different in some ways. $\endgroup$ – Forever Mozart Nov 10 '17 at 21:47
  • $\begingroup$ @ForeverMozart The space is different than the topology. It's the topology on the topology that's the same as the space. Analogous to the double dual of a vector space. Ultrafilters is a topology on the topology. $\endgroup$ – ziggurism Nov 11 '17 at 1:43
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For one example, see the following paper and the references that Lorch cites.

Edgar Raymond Lorch, Continuity and Baire functions, American Mathematical Monthly 78 #7 (August-September 1971), 748-762.

(from bottom of p. 756) In order to penetrate further into this subject it is necessary to give an appropriate structure to $T,$ the set of all coherent topologies. As mentioned earlier, this appropriate structure is itself a topology. This circumstance, that a collection of topologies is topologized, may seem a bit incestuous.

I posted this reference and quote in my 3 June 2002 sci.math post Incestuous Mathematics, where you can find some other examples of a mathematical structure whose underlying set consists of elements each of which has the same (or very similar) type of mathematical structure. See also my 14 September 2006 sci.math post The metric space of metrics. Incidentally, an example that I did not know about back then is Gromov–Hausdorff convergence --- see, for example, Chapter 7 in A Course in Metric Geometry by Burago/Iwanow/Burago (2001).

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