Combining ideas from two of our previous papers ([26]
and [27]), we refine Arhangel’skii Theorem by proving a cardinal inequality
of which this is a special case: any increasing union of strongly
discretely Lindelöf spaces without uncountable free sequences and with
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.
