Star Lindelöfness of products of subspaces of ordinals (Q6547329)

This paper investigates certain properties of topological spaces related to ordinals. The primary focus is on the notions of \(\kappa\)-compactness, star countability, and star Lindelöfness in the context of topological products of subspaces of ordinals.\N\NIn the paper, a space is called \(\kappa\)-compact if every subset with a cardinality greater than or equal to \(\kappa\) has an accumulation point. The concept of star countability is defined for a space \(X\), where for any open cover, there is a countable subset whose star covers the space. Similarly, star Lindelöfness requires a Lindelöf subspace whose star covers the entire space. The authors investigate the conditions under which a product of subspaces of ordinals satisfies these properties.\N\NOne of the key results presented is the characterization of when the product of such spaces is \(\kappa\)-compact. They prove that for a natural number \(n\) and subspaces \(A_i\) of ordinals \(\lambda_i + 1\), the product \(\prod_{i \leq n} A_i\) is \(\kappa\)-compact if and only if certain conditions regarding stationary sets and cofinality are met (Theorem 2.4). Using this result, the authors further demonstrate that a product of subspaces of ordinals is star countable if and only if it is star Lindelöf.\N\NThe implications of this work are relevant in the structure and properties of spaces formed from ordinals, particularly in understanding the interaction between different covering properties and compactness conditions. The paper contributes to a deeper understanding of when a product space is star Lindelöf or star countable based on the extent and compactness properties of the space.