Is there any quasi-compact (= compact, possibly non-Hausdorff) space which is not a quotient of any compact Hausdorff space?
I strongly suspect the answer is yes, yet I couldn't come up with an example so far.
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1$\begingroup$ What about an infinite countable set with the compact topology whose nonempty open subsets are the cofinite subsets? I don't see at the moment whether it's quotient of a Hausdorff compact space. $\endgroup$– YCorCommented Apr 11, 2019 at 19:04
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$\begingroup$ I also suspect most of affine algebraic varieties with Zariski topology would be such examples, but again I have no proof or disproof of this. $\endgroup$– Rick SternbachCommented Apr 11, 2019 at 19:44
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$\begingroup$ Any such space would be compactly generated, maybe this gives some hints. $\endgroup$– Rick SternbachCommented Apr 11, 2019 at 19:46
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$\begingroup$ "Quotient" here has a different definition than the usual one, I guess. It should be a new topology on X, rather than on the set of equivalence classes? $\endgroup$– Nate EldredgeCommented Apr 11, 2019 at 21:40
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3$\begingroup$ Here math.stackexchange.com/a/2794129/94514 is a proof that the one point compactification of the rationals is not a continuous image of (so it is not a quotient of) a compact Hausdorff space. $\endgroup$– Ramiro de la VegaCommented Apr 11, 2019 at 23:25
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