# Article

Full entry | Fulltext not available (moving wall 24 months)
Keywords:
function spaces; $C_p(X,Y)$; Rothberger spaces; $\Psi$-space
Summary:
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 Mrówka 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 $n\in\omega$.
References:
[1] Arhangel'skiĭ A.V.: Topological Function Spaces. Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers, 1992. MR 1144519
[2] Bernal-Santos D.: The Rothberger property on $C_p(X, 2)$. Topology Appl. 196 (2015), 106–119. MR 3422736
[3] Bernal-Santos D., Tamariz-Macarúa Á.: The Menger property on $C_p(X,2)$. Topology Appl. 183 (2015), 110–126. MR 3310340
[4] Hurewicz W.: Über eine Verallgemeinerung des Borelschen Theorems. Math. Z. 24 (1926), 401–421. DOI 10.1007/BF01216792 | MR 1544773
[5] Just W., Miller W., Scheepers M., Szeptycki J.: The combinatorics of open covers II. Topology Appl. 73 (1996), 241–266. DOI 10.1016/S0166-8641(96)00075-2 | MR 1419798
[6] Hrušák M., Szeptycki P.J., Tamariz-Mascarúa Á.: Spaces of functions defined on Mrówka spaces. Topology Appl. 148 (2005), no. (1-3), 239–252. DOI 10.1016/j.topol.2004.09.009 | MR 2118968
[7] Rothberger F.: Eine Verschärfung der Eigenschaft C. Fund. Math. 30 (1938), 50–55.

Partner of