Menger and Rothberger games on convergence spaces

Abstract

We introduce Menger and Rothberger selection principles and games for convergence spaces. Alice plays families that meet every convergent filter, and Bob's selections are required either to retain this property or merely to cover the underlying set. When the convergence is topological, both games recover the classical games. The winning condition requiring an $L$-cover satisfies analogues of the Hurewicz and Pawlikowski characterizations, the condition requiring a cover of $X$ is represented by the weak Menger and Rothberger games. We also show that, for a regular convergence space, a winning strategy for Bob in $\mathsf G_{\fin}(\mathcal C_L,\Cov(X))$ implies an Alster-type covering property. Under hereditary Lindel\"ofness the space is moreover a countable union of compactoid subsets, which are compact in the pretopological case.