# 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 $X\longrightarrow\mathbb{C}$. Recall that $C(X)$ is a vector lattice with the usual order (http://planetmath.org/PartialOrder): $f\leq g\Longleftrightarrow g-f$ takes positive (or zero) values.

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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 CompactHausdorffSpaceIsExtremallyDisconnectedIfItsFunctionAlgebraIsABoundedCompleteLattice 2013-03-22 17:53:08 2013-03-22 17:53:08 asteroid (17536) asteroid (17536) 5 asteroid (17536) Theorem msc 54G05 msc 46J10 msc 06F20 sufficient condition for a compact Hausdorf space to be extremally disconnected