Monotone Insertion of Semi-Continuous Functions on Stratifiable Spaces

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) vertical bar h vertical bar <= h'; (ii) h(1)' <= h(2)' whenever vertical bar h(1)vertical bar <= vertical bar h(2)vertical bar. (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* such that (i) h' is locally bounded at each x is an element of U-h with respect to R, where U-h = {x is an element of X : h is locally bounded at x with respect to R g; (ii) vertical bar h vertical bar <= h'; (iii) h(1)' <= h(2)' whenever h(1)vertical bar <= vertical bar h(2)vertical bar. We give a negative answer to the problem posed by K.D. Li [14].