= Finite-fiber compact extension
{title2=$X=Y\times\{1,\ldots,q\}$}
For a skew-product extension with finitely many labeled fibers, the orbit of a bounded observable on each fiber lies in a fixed bounded subset of a finite-dimensional vector space. A finite net there supplies global centers constant in the base coordinate. Bounded observables are dense in $L^2$, so this is a <compact extension>. Global almost periodicity can still fail because the base can be weakly mixing.
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