Cardinal inequalities involving the weak Rothberger and cellularity games

We prove several results in the theory of topological cardinal invariants involving the game-theoretic versions of the weak Lindelöf degree and of cellularity. One of them is related to Bell, Ginsburg and Woods’s 1978 question of whether every weakly Lindelöf regular first-countable space has cardinality at most continuum and another one is connected with Arhangel’skii’s 1970 question on the weak Lindelöf degree of the \(G_\delta \) $G_{\delta }$ topology on a compact space. We provide a few application of our results, including some bounds on the cardinality of sequential and radial spaces. We finish with a series of counterexamples, which show the sharpness of our results and disprove a few natural conjectures about the impact of infinite games on topological cardinal invariants.