On the cardinality of Lindelöf spaces with points Gδ

We investigate Arhangelʹskǐiʹs problem of whether Lindelöf T2 spaces with points Gδ have cardinality ⩽ 2ℵ0. We extend results of Shelah, using supercompact cardinals to exclude cardinalities between ℵ1 and 2ℵ1. We also obtain several results concerning the possible structure of large Lindelöf spaces with points Gδ.