Countable tightness and the Grothendieck property in \(C_p\)-theory (Q2105061)

The paper addresses various open problems concerning the Grothendieck property. A space \(X\) is a \textit{\(g\)-space} if countably compact subsets in \(X\) have compact closures; furthermore, \(X\) is a \textit{Grothendieck space} if the space of continuous real-valued functions with the pointwise topology \(C_p(X)\) is a hereditary g-space. Answering questions of Arhangel'skiĭ it is proven to be undecidable whether countably tight spaces with Lindelöf finite powers are Grothendieck; moreover, it is shown that PFA implies that Lindelöf countably tight spaces are Grothendieck. The author also proves that various other consequences of MA\(_{\omega_1}\) and PFA are not theorems of ZFC.