Lindelof indestructibility, topological games and selection principles

Arhangel'skii proved that if a first countable Hausdorff space is Lindel\"of, then its cardinality is at most $2^{\aleph_0}$. Such a clean upper bound for Lindel\"of spaces in the larger class of spaces whose points are ${\sf G}_{\delta}$ has been more elusive. In this paper we continue the agenda started in F.D. Tall, On the cardinality of Lindel\"of spaces with points $G_{\delta}$, Topology and its Applications 63 (1995), 21 - 38, of considering the cardinality problem for spaces satisfying stronger versions of the Lindel\"of property. Infinite games and selection principles, especially the Rothberger property, are essential tools in our investigations