In this paper $LJ$-spaces are introduced and studied. They are a common generalization of Lindelöf spaces and $J$-spaces researched by E. Michael. A space $X$ is called an $LJ$-space if, whenever $\lbrace A,B\rbrace $ is a closed cover of $X$ with $A\cap B$ compact, then $A$ or $B$ is Lindelöf. Semi-strong $LJ$-spaces and strong $LJ$-spaces are also defined and investigated. It is demonstrated that the three spaces are different and have interesting properties and behaviors.
A space $X$ is $\mathcal L$-starcompact if for every open cover $\mathcal U$ of $X,$ there exists a Lindelöf subset $L$ of $X$ such that $\mathop {\mathrm St}(L,{\mathcal U})=X.$ We clarify the relations between ${\mathcal L}$-starcompact spaces and other related spaces and investigate topological properties of ${\mathcal L}$-starcompact spaces. A question of Hiremath is answered.
Let P be a topological property. A space X is said to be star P if whenever U is an open cover of X, there exists a subspace A ⊆ X with property P such that X = St(A,U), where St(A, U) = ∪ {U ∈ U : U ∩A ≠ ∅}. In this paper, we study the relationships of star P properties for P ∈ {Lindelöf, compact, countably compact} in pseudocompact spaces by giving some examples.
In this paper, we prove the following statements: (1) For every regular uncountable cardinal $\kappa $, there exist a Tychonoff space $X$ and $Y$ a subspace of $X$ such that $Y$ is both relatively absolute star-Lindelöf and relative property (a) in $X$ and $e(Y,X) \ge \kappa $, but $Y$ is not strongly relative star-Lindelöf in $X$ and $X$ is not star-Lindelöf. (2) There exist a Tychonoff space $X$ and a subspace $Y$ of $X$ such that $Y$ is strongly relative star-Lindelöf in $X$ (hence, relative star-Lindelöf), but $Y$ is not absolutely relative star-Lindelöf in $X$.