Recall that a sober space is a topological space such that every irreducible closed subset is the closure of exactly one point.

Is there any area of mathematics outside of general topology where non-sober spaces are objects of interest?

I can't think of any cases at least in algebraic topology where such a space can't be replaced by a homotopy-equivalent sober space. All manifolds are sober, and all schemes are sober, so it seems like as far as geometry goes, all spaces are also sober.

I suspect that if non-sober spaces show up, it might be in some special cases in infinite-dimensional functional analysis, but that's just a guess.

PS - Community Wiki It!

PPS - The question is motivated by the comment on this page in Johnstone's Elephant:

Quoted page of Johnstone

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    $\begingroup$ I’m posting as a comment, since this is a might-be-example, rather than an example, but I believe that it’s not known whether the Ziegler spectrum of a ring is always sober. $\endgroup$ – Jeremy Rickard Jan 10 at 17:01
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    $\begingroup$ @JeremyRickard Wikipedia says they frequently fail to be T0, and sober implies T0. Ergo, your example is indeed an example. $\endgroup$ – Harry Gindi Jan 10 at 17:06
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    $\begingroup$ Hmm, Wikipedia also says that it’s not known whether they’re always sober, so this line of argument may be proving more about Wikipedia than about Ziegler spectra. $\endgroup$ – Jeremy Rickard Jan 10 at 17:11
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    $\begingroup$ @DimaPasechnik Yeah, as a classical variety, but it's in a non-essential way. The schematic Zariski topology makes it correct. $\endgroup$ – Harry Gindi Jan 10 at 17:29
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    $\begingroup$ “Soberfied”? “Soberification”? Clearly the terminology should derive from “sober up”! $\endgroup$ – Jeremy Rickard Jan 10 at 18:26

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