Remarks on semi-Menger and star semi-Menger spaces (Q2107422)

This paper proves that for an extremally disconnected \(S\)-paracompact \(T_{2}\) space, the properties of semi-Menger, Menger, strongly star semi-Menger, strongly star-Menger, star semi-Menger, star-Menger, almost semi-Menger, almost Menger, almost star semi-Menger, almost star-Menger are equivalent. Also the authors show that a weakly semi-continuous (strongly \(\theta\)-semi-continuous) image of a semi-Menger (almost semi-Menger) space is almost semi-Menger (semi-Menger) and give characterizations of semi-Menger and star semi-Menger spaces. The main result is as follows: Corollary 5.7. For an extremally disconnected \(S\)-paracompact \(T_{2}\) space \(X\), the following statements are equivalent: \begin{itemize} \item[1.] \(X\) is almost-Menger; \item[2.] \(X\) is almost semi-Menger; \item[3.] \(X\) is semi-Menger; \item[4.] \(X\) is Menger; \item[5.] \(X\) is strongly star-Menger; \item[6.] \(X\) is strongly star semi-Menger; \item[7.] \(X\) is star semi-Menger; \item[8.] \(X\) is almost star semi-Menger; \item[9.] \(X\) is almost star-Menger; \item[10.] \(X\) is star-Menger. \end{itemize}