General topology Compactifications Theorem

Gleason's theorem on projective compact spaces

Builds on Compact space Hausdorff space Continuous map Extremally disconnected space

Statement

A compact Hausdorff space $P$ is projective if it satisfies the following lifting property. For all compact Hausdorff spaces $Y$ and $Z$ , every continuous surjection $f \colon Y \to Z$ and every continuous map $g \colon P \to Z$ , there is a continuous map $h \colon P \to Y$ with $f \circ h = g$ .

Let $P$ be a compact Hausdorff space. Then $P$ is projective if and only if $P$ is extremally disconnected.

Remarks

For one direction, let $D$ be the set $P$ with the discrete topology. The identity $D \to P$ extends to a surjection $\beta D \to P$ from the Stone–Čech compactification. If $P$ is projective, it is a retract of $\beta D$ , which is extremally disconnected, and retracts inherit the property. The other direction uses a closed subset of the pullback that is minimal among those mapping onto $P$ , found by Zorn's lemma.

Example at work: $\beta \N$ is extremally disconnected, so it is projective.

Going deeper

For a student

Every compact Hausdorff space $X$ is a continuous image of a projective one, namely $\beta D$ for $D$ the set $X$ with the discrete topology. In fact $X$ has a projective cover, often called its Gleason cover. Through Stone duality, extremally disconnected compact Hausdorff spaces correspond to complete Boolean algebras.

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Compact space concept
Continuous map concept
Extremally disconnected space concept
Hausdorff space concept
Gleason's theorem on projective compact spaces statement

Tag	`CPZKZ2AV0G`
URI	`m2://CPZKZ2AV0G`
Kind	statement, theorem
Writing	W0 stub
MSC	54G05, 18G05, 54D30
Status	This copy of the site has no build data. Run `m2 status CPZKZ2AV0G`.
About	Extremally disconnected space
Uses	Compact space, Hausdorff space, Continuous map, Extremally disconnected space
Changed	2 November 2026

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