Combinatorics of Large Covers

Recall that a large cover of a topological space is a cover in which every point is covered by infinitely many members of the cover. Expanding on results of Scheepers from the 90s, we will show that the Menger game is perfect- and limited-information equivalent to what we call the λ-Menger game, where the first player chooses large covers and the second player is attempting to form a large cover with their finite selections. We will also discuss the single-selection analog and note that, in the λ-Rothberger game in the context of T_1 spaces, the second player having a Markov winning strategy on a space is equivalent to the space having a countable topology. Consequently, the Rothberger game and the λ-Rothberger game are not equivalent in the same way that the Menger analogues are.