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We show that the Gromov-Hausdorff limit of a sequence of leaves in a compactfoliation is a covering space of the limiting leaf which is no larger than thisleaf's holonomy cover. We also show that convergence to such a limit is smoothinstead of merely Gromov-Hausdorff. Corollaries include Reeb's local stabilitytheorem, part of Epstein's local structure theorem for foliations by compactleaves, and a continuity theorem of Alvarez and Candel. Several examples arediscussed.

Reeb stability and the Gromov-Hausdorff limits of leaves in compact foliations