Monotone insertion of semi-continuous maps to ordered topological vector spaces

In this paper, we consider the problem of inserting semi-continuous maps with values into some ordered topological vector spaces in a monotone fashion. We answer the questions of K. Yamazaki and provide a new characterization of stratifiable spaces, semi-stratifiable spaces, countably paracompact spaces, countably metacompact spaces, perfect spaces, and perfectly normal spaces during this process.