Some weak versions of the -spaces

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 ) ⩽ 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 σ- m 3 -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.