# "weakly functorial resolution" of quasi-compact T_1 topological space by quasi-compact Hausdorff space

I have an arguably weird question: Let $$X$$ be a quasi-compact $$T_1$$ topological space, could there be a construction that takes such an $$X$$ as input and outputs a surjection $$X' \to X$$ with the following two additional properties:

1. $$X'$$ is quasi-compact and Hausdorff;
2. The cardinality of $$X'$$ is not too much bigger than that of $$X$$, say something like $$|X'| < 2^{2^{2^{|X|}}}$$ would suffice;
3. the association $$X \mapsto (X' \to X)$$ is "weakly functorial" in the sense that once we are given a map $$S \to X$$ between quasi-compact $$T_1$$ topological spaces, we may produce a commutative diagram:$$\require{AMScd}\begin{CD} S' @>>> X' \\ @VVV @VVV \\ S @>>> X \end{CD}$$

Note that the Stone-Cech compactification doesn't work since $$X$$ is not Hausdorff, and Hausdorf quotient of $$X$$ is going the wrong direction.

The motivation for this is to give an alternative proof of Clausen-Scholze's lecture notes on Condensed Mathematics Proposition 2.15 (first half): The statement there, roughly speaking, is that given any quasi-compact $$T_1$$ topological space $$X$$ which satisfies some cardinality bound $$|X| < \kappa$$ for certain kind of strong limit cardinal $$\kappa$$, and suppose we are given some map $$f: S \to X$$ for some extremally disconnected set $$S$$, then one can factorize $$f$$ as $$S \to X' \to X$$ where $$X'$$ is a quasi-compact Hausdorff space with $$|X'| < \kappa$$ as well.

If the above "weakly functorial resolution" exists, then one can prove the said Proposition rather easily. (The current proof is indeed very short, except I can't quite understand it...)

• Simple diagrams on MO can be made with amscd. ctan.org/pkg/amscd. I have added it to your question. Oct 14, 2023 at 11:24