  Combining ideas from two of our previous papers, we refine Arhangel'skii
Theorem by proving a cardinal inequality of which this is a special case: any
increasing union of strongly discretely Lindelof spaces with countable free
sequences and countable pseudocharacter has cardinality at most continuum. We
then give a partial positive answer to a problem of Alan Dow on reflection of
cardinality by closures of discrete sets.
