On cellular-compact and related spaces

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.