Remarks on set selectively star-ccc spaces (Q2673854)

Recall that for \(k\in\mathbb N\) a space \(X\) is \textit{set selectively \(k\)-star-ccc} if for each non-empty \(B\subset X\), every open cover \(\mathcal U\) of \(\overline{B}\) and every sequence \(\langle\mathcal A_n\rangle\) of maximal cellular open families in \(X\) there is a sequence \(\langle A_n\rangle\) such that \(B\subset \mathrm{St}^k(\bigcup_nA_n,\mathcal U)\) and \(A_n\in\mathcal A_n\) for each \(n\in\mathbb N\). It is shown that set selectively star-ccc is preserved under open F\(_\sigma\) subsets. On the other hand examples are given of a Tychonoff pseudocompact selectively 2-star-ccc space that is not selectively \(k\)-star-ccc, a Tychonoff set selectively star-ccc space having a regular-closed G\(_\delta\)-subspace which is not set selectively star-ccc and, under \(2^{\aleph_0}=2^{\aleph_1}\), a normal selectively 2-star-ccc space that is neither strongly star-Lindelöf nor set selectively star-ccc.