Star Lindelöfness of Products of Subspaces of Ordinals

Abstract

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 $\operatorname{St}(F, \mathcal{U})=X$. In this paper, we give a characterization when the product $\prod_{i \leq n} A_i$ is $\kappa$-compact, where $\kappa>\omega$ is regular, $n$ is a natural number, and each $A_i$ is a subspace of an ordinal $\lambda_i+1$. By using this result, we show that such a product $\prod_{i \leq n} A_i$ is star countable if and only if it is star Lindelöf.