The Stacks project

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

  1. $X$ is extremally disconnected,

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

  3. 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.

Proof. It is clear that (3) implies (2). On the other hand, if (2) holds and $X \to Z$ and $Y \to Z$ are as in (3), then (2) assures there is a section to the projection $X \times _ Z Y \to X$ which implies a suitable dotted arrow exists (details omitted). Thus (3) is equivalent to (2).

Assume $X$ is extremally disconnected and let $f : Y \to X$ be as in (2). By Lemma 5.26.5 there exists a quasi-compact subset $E \subset Y$ such that $f(E) = X$ but $f(E') \not= X$ for all proper closed subsets $E' \subset E$. By Lemma 5.26.4 we find that $f|_ E : E \to X$ is a homeomorphism, the inverse of which gives the desired section.

Assume (2). Let $U \subset X$ be open with complement $Z$. Consider the continuous surjection $f : \overline{U} \amalg Z \to X$. Let $\sigma $ be a section. Then $\overline{U} = \sigma ^{-1}(\overline{U})$ is open. Thus $X$ is extremally disconnected. $\square$


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