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Hyperstonean locales can be defined as Stonean locales that admit sufficiently many normal valuations.
Assuming the axiom of choice, Stonean locales are spatial, so the category of hyperstonean locales is equivalent to the category of hyperstonean spaces.
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Given a compact Hausdorff space $X$, its hyperstonean cover is defined as a continuous map of compact Hausdorff spaces
$hX \to X$that under the Gelfand duality corresponds to the canonical inclusion $C(X)\to C(X)^{**}$ into the double dual.
The hyperstonean cover of a compact Hausdorff space is introduced in
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