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We consider the question of whether a compact space will always have a discrete subset whose closure has the same cardinality as the whole space. We obtain many positive results for compact spaces of countable tightness and a consistent negative result for a space of tightness and density ?1.

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Eric van Douwen [5] produced a maximal crowded extremally disconnected regular space and showed that its Čech-Stone compactification is an at most two-to-one image of β ℕ. We construct for any n ≧ 3 an example of a compact crowded space X n that is an image of β ℕ under a map all of whose fibers have either size n or n − 1. We also show that under CH this is best possible.

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