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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.

I strongly suspect the answer is yes, yet I couldn't come up with an example so far.

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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Is there any quasi-compact 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.