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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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.

单词	CompactHausdorffSpaceIsExtremallyDisconnectedIfItsFunctionAlgebraIsABoundedCompleteLattice
释义	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. $\\,$ 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.
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