Nearly \(\mu\)-Lindelöfness via hereditary class (Q6629716)

In this paper, the authors define and study the notion of hereditary class on nearly $\mu$-Lindelöf spaces. Moreover, they study the effects of some types of continuity of hereditary class on nearly $\mu$-Lindelöf space by properties of the function. Also, more variations between these spaces and some known spaces are investigated.\N\NThe authors prove the following results.\N\N\textbf{Theorem 2.2.} If $X$ is a $\mathcal{N}\mu\mathcal{H}$-Lindelöf space, then $X$ is a $\mathcal{W}\mu\mathcal{H}$-Lindelöf space.\N\N\textbf{Theorem 2.4.} Let $X_{\mu}$ be a $\mu$-regular space. Then, the following are equivalent:\N\begin{itemize}\N\item[(1)] $(X_{\mu}, \mathcal{H})$ is $\mathcal{N}\mu\mathcal{H}$-Lindelöf;\N\item[(2)] $(X_{\mu}, \mathcal{H})$ is $\mu\mathcal{H}$-Lindelöf;\N\item[(3)] $(X_{\mu}, \mathcal{H})$ is $\mathcal{W}\mu\mathcal{H}$-Lindelöf.\N\end{itemize}\N\N\textbf{Theorem 2.7.} Let $A$ be a subset of $X_{\mu}$. The following statements are equivalent:\N\begin{itemize}\N\item[(1)] $A$ is $\mathcal{N}\mu\mathcal{H}$-Lindelöf;\N\item[(2)] For any collection $\mathcal{F}=\{U_{\lambda}:\lambda\in\Lambda\}$ of a $\mu$-closed subset of $X$ such that $[\bigcap\{U_{\lambda}:\lambda\in\Lambda\}]\cap A=\emptyset$, there exists a countable sub-collection $\{U_{\lambda}:\lambda\in\Lambda_{0}\subseteq\Lambda\}$ of $\mathcal{F}$ such that \N\[\N[\bigcap_{\lambda\in\Lambda_{0}}\mathrm{Cl}_{\mu}\mathrm{Int}_{\mu}(U_{\lambda})]\bigcap A\in\mathcal{H};\N\]\N\item[(3)] For any collection $\mathcal{F}=\{U_{\lambda}:\lambda\in\Lambda\}$ of a $\mu$-regular closed subset of $X$ such that $[\bigcap\{U_{\lambda}:\lambda\in\Lambda\}]\cap A=\emptyset$, there exists a countable sub-collection $\{U_{\lambda}:\lambda\in\Lambda_{0}\subseteq\Lambda\}$ of $\mathcal{F}$ such that \N\[\N[\bigcap_{\lambda\in\Lambda_{0}}\mathrm{Cl}_{\mu}\mathrm{Int}_{\mu}(U_{\lambda})]\cap A\in\mathcal{H}.\N\]\N\end{itemize}