Monotone Insertion of Semi-Continuous Functions on Stratifiable Spaces

Ying-Ying Jin, Li-Hong Xie, Hong-Wei Yue

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: \begin{itemize} ıem[(1)] A space $X$ is $k$-MCM if and only if for each locally bounded real-valued function $h\colon X\to\mathbb R$, there exists a lower semi-continuous and $k$-upper semi-continuous function $h'\colon\mathbb R$ such that (i) $|h|\leq h'$; (ii) $h'_1\leq h'_2$ whenever $|h_1|\leq|h_2|$. ıem[(2)] A space $X$ is stratifiable if and only if for each function $h\colon X\to\mathbb R^*$ ($\mathbb R^*$ is the generalized real number set), there is a lower semi-continuous function $h'\colon X\to\mathbb R^*$ such that (i) $h'$ is locally bounded at each $x\in U_h$ with respect to $\mathbb R$, where $U_h=\{x\in X:h\text{ is locally bounded at }x\text{ with respect to }\mathbb R\}$; (ii) $|h|\leq h'$; (iii) $h'_1\leq h'_2$ whenever $|h'_1|\leq|h'_2$|. \end{itemize} We give a negative answer to the problem posed by K.\,D. Li [14].