Monotone insertion of semi-continuous functions on stratifiable spaces

Abstract

In this paper, we consider the problem of inserting semi-continuous function above the (generalized) real-valued function in a monotone fashion. We provide some characterizations of stratifiable spaces, semi-stratifiable spaces, and k-monotonically countably metacompact spaces (k-MCM) and so on. It is established that: (1) A space X is k-MCM if and only if for each locally bounded real-valued function h : X ? R, there exists a lower semi-continuous and k-upper semi-continuous function h': X ? R such that (i) |h|? h', (ii) h'1 ? h'2 whenever |h1| ? |h2|. (2) A space X is stratifiable if and only if for each function h : X ? R* (R* is the generalized real number set), there is a lower semi-continuous function h' : X ? R