Some weak versions of the M1-spaces

We mainly introduce some weak versions of the M1-spaces, and study some properties about these spaces. The mainly results are that: (1) If X is a compact scattered space and i(X)⩽3, then X is an s-m1-space; (2) If X is a strongly monotonically normal space, then X is an s-m2-space; (3) If X is a σ-m3-space, then t(X)⩽c(X), which extends a result of P.M. Gartside (1997) in [7]. Moreover, some questions are posed in the paper.