Star countability of products of subspaces of ordinals (Q6551731)

For an infinite cardinal \(\kappa\), a topological space \(X\) is called \(\kappa\)-compact if every \(F \subseteq X\) with \(|F| \geq \kappa\) has an accumulation point. A space \(X\) is said to be star countable (respectively star Lindelöf) if for every open cover \(\mathcal{U}\) of \(X\), there exists a countable subset (respectively a Lindelöf subspace) \(F\) of \(X\) such that \(\operatorname{St}(F, \mathcal{U})=X\).\N\NIn this paper, the authors give two main results.\N\NThe first one is a characterization when \(A \times B\) is \(\kappa\)-compact for subspaces \(A\) and \(B\) of an ordinal \(\lambda\), where \(\kappa>\omega\) is a regular cardinal.\N\NThe second one shows that for subspaces \(A\) and \(B\) of an ordinal, star countable, star Lindelöf, and having countable extent are equivalent properties in the space \(A \times B\).