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:

Clearlythe terminology should derive from “sober up”! $\endgroup$ – Jeremy Rickard Jan 10 at 18:26