Strongly τ-pseudocompact spaces

A new notion of a strongly τ-pseudocompact space is introduced, and its relation to the notion of initially τ-compact space is studied. A theorem of J.F. Kennison (Kennison, 1962) is generalized to this case, which leads to new interesting questions. In particular, it is proved that every strongly 2ω-pseudocompact space of countable tightness is compact, and that every strongly 2ω-pseudocompact space is initially τ+-compact. It is also established that every ω1-pseudocompact topological group of countable tightness is metrizable and, therefore, compact. Several open problems are formulated.