Proposition 5.26.6. Let $X$ be a Hausdorff, quasi-compact topological space. The following are equivalent

$X$ is extremally disconnected,

for any surjective continuous map $f : Y \to X$ with $Y$ Hausdorff quasi-compact there exists a continuous section, and

for any solid commutative diagram

\[ \xymatrix{ & Y \ar[d] \\ X \ar@{..>}[ru] \ar[r] & Z } \]of continuous maps of quasi-compact Hausdorff spaces with $Y \to Z$ surjective, there is a dotted arrow in the category of topological spaces making the diagram commute.

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