Dowker-type example and Arhangelskii's \(\alpha_2\)-property (Q386849)

For a topological space \(X\), let \(C_{p}(X)\) be the space of all continuous real-valued functions on \(X\) endowed with the topology of pointwise convergence and let \(\Gamma\) (resp. \(\Gamma_{cl}\)) be the family of all open (resp. clopen) point-cofinite covers of \(X\). In the paper under review, the authors give a normal zero-dimensional space \(X\) satisfying \(\mathcal{S}_{1}(\Gamma_{cl},\Gamma_{cl})\) but not \(\mathcal{S}_{1}(\Gamma,\Gamma)\), answering by this in the negative a question of M. Sakai and partially answering a question of M. Scheepers. They also show that, for this space, \(C_{p}(X,2)\) is an \(\alpha_{2}\)-space but \(C_{p}(X)\) is not an \(\alpha_{2}\)-space.