Some weak versions of the $M_1$-spaces

Abstract

We mainly introduce some weak versions of the $M_1$-spaces, and study some properties about these spaces. The mainly results are that: (1) If $X$ is a compact scattered space and $i(X)\le 3$, then $X$ is an $s$-$m_1$-space; (2) If $X$ is a strongly monotonically normal space, then $X$ is an $s$-$m_2$-space; (3) If $X$ is a $\sigma$-$m_3$ space, then $t(X)\le c(X)$, which extends a result of P.M. Gartside in \cite{CP}. Moreover, some questions are posed in the paper.

8 pages