Some weaker monotone separation and basis properties

Ito showed that stratifiable spaces in which there is a closure preserving local base at each point are M1. That local version of M1 is called m1, while m2 and m3 are analogously defined. We look at some of the properties of these spaces. A subtler weakening of one of the characterizations of monotone normality leads to a new class of spaces we call “monotonically T2”. Besides giving a considerably larger class of spaces, the monotone T2 property is, for example, preserved under arbitrary box products. Also, there is a strong relationship between the mi-spaces and the monotone T2 property, and in some circumstances they are equivalent. We also discuss similar analogs of acyclic and strong monotone normality.