Abstract
We solve a problem posed by Tkachuk and Wilson, Question 5.10 in [16], on whether every first countable cellular-compact space is weakly Lindelöf. We actually obtain a stronger result and, as a by-product of it, we present a somewhat different proof of Tkachuk and Wilson theorem on the cardinality of a first countable cellular-compact space, valid for the wider class of Urysohn spaces. Moreover, our result holds for a class of spaces in between cellular-compact and cellular-Lindelöf. We conclude with some comments on the cardinality of a weakly linearly Lindelöf space.
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