Abstract:In this paper, we prove the following statements: (1) For any cardinal $\kappa$, there exists a Tychonoff star-Lindel\"of space $X$ such that $a(X)\geq \kappa$. (2) There is a Tychonoff discretely star-Lindel\"of space $X$ such that $aa(X)$ does not exist. (3) For any cardinal $\kappa$, there exists a Tychonoff pseudocompact $\sigma$-starcompact space $X$ such that $\operatorname{st}\text{-}l(X)\geq \kappa$.
Keywords: star-Lindel\"of number, the Aquaro number, the absolute Aquaro number, star-Lindel\"of, centered-Lindel\"of, discretely star-Lindel\"of, absolutely discretely star-Lindel\"of, $\sigma$-starcompact, pseudocompact
AMS Subject Classification: 54A25 54D20