Not realcompact images of not Lindelöf spaces (Q1329763)

It is a question of \textit{S. Mrówka} [General Topology Relations modern Analysis Algebra, Proc. Kanpur topol. Conf. 1968, 207-214 (1971; Zbl 0229.54024)] whether every non-Lindelöf Tikhonov space admits a non- realcompact Tikhonov continuous image. The authors answer this question affirmatively for spaces \(X\) which in addition satisfy any one of these conditions: (a) the Lindelöf degree of \(X\) is \(\leq 2^ \omega\); (b) \(X\) has countable tightness; (c) \(X\) maps continuously onto \(X^ 2\). They show also: Any space \(X\) with an uncountable closed discrete subspace has a one-to-one non-realcompact continuous image. Several unsolved problems are posed, including Mrowka's question in its original form.