Infinite games and cardinal properties of topological spaces

Inspired by work of Scheepers and Tall, we use properties de-fined by topological games to provide bounds for the cardinality of topological spaces. We obtain a partial answer to an old question of Bell, Ginsburg and Woods regarding the cardinality of weakly Lindelöf first-countable regular spaces and answer a question recently asked by Babinkostova, Pansera and Scheepers. In the second part of the paper we study a game-theoretic variant of the countable chain condition, introduced by Daniels, Kunen and Zhou in 1994. We prove that it is equivalent to countable π-weight for spaces having countable local π-weight at every point, and exploit it to give a new proof of Shapirovskii's classical bound on the number of regular open sets of a regular topological space.