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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