Abstract
The Grothendieck property has become important in research on the definability of pathological Banach spaces [4], [9], and especially [10]. We here answer a question of Arhangel'skiĭ by proving it undecidable whether countably tight spaces with Lindelöf finite powers are Grothendieck. We answer another of his questions by proving that PFA implies Lindelöf countably tight spaces are Grothendieck. We also prove that various other consequences of MAω1 and PFA considered by Arhangel'skiĭ, Okunev, and Reznichenko are not theorems of ZFC.
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