Countable Tightness and the Grothendieck Property in $C_p$-Theory

TL;DR

This paper investigates the conditions under which countably tight spaces with Lindelöf finite powers are Grothendieck, showing some results depend on additional set-theoretic axioms beyond ZFC.

Contribution

It proves the undecidability of whether such spaces are Grothendieck within ZFC and demonstrates that certain implications of set-theoretic axioms are not provable in ZFC.

Findings

Undecidability of the Grothendieck property for certain spaces in ZFC.

PFA implies Lindelöf countably tight spaces are Grothendieck.

Some consequences of MA_ω1 and PFA are independent of ZFC.

Abstract

The Grothendieck property has become important in research on the definability of pathological Banach spaces [CI], [HT], and especially [HT20]. We here answer a question of Arhangel'ski\u{\i} by proving it undecidable whether countably tight spaces with Lindel\"of finite powers are Grothendieck. We answer another of his questions by proving that $\mathrm{PFA}$ implies Lindel\"of countably tight spaces are Grothendieck. We also prove that various other consequences of $\mathrm{MA}_{\omega_1}$ and $\mathrm{PFA}$ considered by Arhangel'ski\u{\i}, Okunev, and Reznichenko are not theorems of $\mathrm{ZFC}$.