Closures of discrete sets often reflect global properties
Ofelia T. Alas, Vladimir V. Tkachuk and and Richard G. Wilson
We deal with spaces X in which the closure of every discrete
subspace has a given property.
We prove that, in many cases, the space X has the same property.
In particular, if Pin {perfect normality,
tightness <= \kappa, character <= \kappa, countability},
then a compact space X has P if and only if [`D]
has P for every discrete D subset X. We also establish that,
under MA+ not CH, if X is compact and the closure of any
discrete subspace of X is metrizable, then X is metrizable.
On the other hand, under CH, there exists a compact
non-metrizable space X in which the closure of any discrete subset
is metrizable.
