The Rothberger property on $C_p(\Psi(\mathcal A),2)$

Abstract

A space $X$ is said to have the Rothberger property (or simply $X$ is Rothberger) if for every sequence $\langle\,\mathcal U_n:n\in \omega\,\rangle$ of open covers of $X$, there exists $U_n\in \mathcal U_n$ for each $n\in\omega$ such that $X = \bigcup_{n\in \omega}U_n$. For any $n\in \omega$, necessary and sufficient conditions are obtained for $C_p(\Psi(\mathcal A),2)^n$ to have the Rothberger property when $\mathcal A$ is a Mrowka mad family and, assuming CH (the Continuum Hypothesis), we prove the existence of a maximal almost disjoint family $\mathcal A$ for which the space $C_p(\Psi(\mathcal A),2)^n\,$ is Rothberger for all $n\in\omega$.