English

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

General Topology 2020-07-20 v1

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 PFA\mathrm{PFA} implies Lindel\"of countably tight spaces are Grothendieck. We also prove that various other consequences of MAω1\mathrm{MA}_{\omega_1} and PFA\mathrm{PFA} considered by Arhangel'ski\u{\i}, Okunev, and Reznichenko are not theorems of ZFC\mathrm{ZFC}.

Keywords

Cite

@article{arxiv.2007.08579,
  title  = {Countable Tightness and the Grothendieck Property in $C_p$-Theory},
  author = {Franklin D. Tall},
  journal= {arXiv preprint arXiv:2007.08579},
  year   = {2020}
}