Some alternative interpretations of strongly star semi-Rothberger and related spaces (Q6629717)

In this article, the authors consider some under-appreciated characteristics of strongly star-Rothberger subsets. With the aid of the SSI property and MSSI property, semi-Rothberger and star semi-Rothberger spaces are represented by families of closed sets. Finally, they produce several selection principle like attributes that can reflect the previously mentioned sequential covering features in an inverted form.\N\NIn particular, the authors prove the following results.\N\NProposition 3.1. The union of two semi-Rothberger sets is semi-Menger.\N\NTheorem 3.4. The following statements are equivalent:\N\begin{itemize}\N\item[(1)] \(X\) is semi-Rothberger.\N\item[(2)] For every sequence \(\{\mathcal{F}_{n}: n\in\mathbb{N}\}\) of the family of semi-closed sets in \(X\) having SSI property, there exists \(n_{0}\in\mathbb{N}\) such that \(\bigcap\mathcal{F}_{n_{0}}\neq\emptyset\).\N\end{itemize}\N\NTheorem 3.5. The following statements are equivalent:\N\begin{itemize}\N\item[(1)] \(X\) is semi-Rothberger.\N\item[(2)] The selection principle \(S_{1}(\mathrm{SO}(X), \mathrm{SO}(X))\) holds for the space \(X\).\N\item[(3)] The selection principle \(S_{1}(\mathrm{SC}(X), \mathrm{SC}(X))\) holds for the space \(X\).\N\end{itemize}\N\NProposition 4.1. The union of two strongly star semi-Rothberger subsets is a strongly star semi-Menger subset.\N\NTheorem 4.4. The following statements are equivalent:\N\begin{itemize}\N\item[(1)] \(X\) is strongly star semi-Rothberger.\N\item[(2)] If a sequence of closed sets \(\{\mathcal{F}_{n}: n\in\mathbb{N}\}\) in \(X\) have MSSIP property, then there exists \(n_{0}\in\mathbb{N}\) such that \(\bigcap\mathcal{F}_{n_{0}}\neq\emptyset\).\N\end{itemize}