compact Hausdorff space is extremally disconnected if its function algebra is a bounded complete lattice

Let $X$ be a compact Hausdorff space and $C(X)$ the algebra of continuous functions $X⟶ℂ$. Recall that $C(X)$ is a vector lattice with the usual order (http://planetmath.org/PartialOrder): $f\le g⟺g-f$ takes positive (or zero) values.

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Theorem - If every subset of $C(X)$ that is bounded from above has a least upper bound (i.e. $C(X)$ is a bounded complete lattice), then $X$ is extremally disconnected.

Title	compact Hausdorff space is extremally disconnected if its function algebra is a bounded complete lattice
Canonical name	CompactHausdorffSpaceIsExtremallyDisconnectedIfItsFunctionAlgebraIsABoundedCompleteLattice
Date of creation	2013-03-22 17:53:08
Last modified on	2013-03-22 17:53:08
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	5
Author	asteroid (17536)
Entry type	Theorem
Classification	msc 54G05
Classification	msc 46J10
Classification	msc 06F20
Synonym	sufficient condition for a compact Hausdorf space to be extremally disconnected