Daniel Bernal-Santos
The Rothberger property on $C_p(\Psi(\mathcal A),2)$

Comment.Math.Univ.Carolin. 57,1 (2016) 83-88.

Abstract:A space $X$ is said to have the {\it Rothberger property\/} (or simply $X$ is {\it 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 Mr\'owka 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 $n\in\omega$.

Keywords: function spaces; $C_p(X,Y)$; Rothberger spaces; $\Psi$-space

DOI: DOI 10.14712/1213-7243.2015.145
AMS Subject Classification: 54C35 54D35 03G10 54D45 54C45