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:
\begin{enumerate}

\item A space $X$ is $k$-MCM if and only if for each locally bounded
real-valued function $h: X\rightarrow \mathbb{R}$,
there exists a lower semi-continuous and $k$-upper semi-continuous function $h': X \rightarrow \mathbb{R}$ such that
(i) $ | h| \leq h' $;
(ii) $h'_1 \leq h'_2$ whenever $| h_1| \leq | h_2|$.

\item A space $X$ is stratifiable
if and only if for each function $h: X\rightarrow \mathbb{R}^*$ ($\mathbb{R}^*$ is the generalized real number set),
there is a lower semi-continuous function $h': X\rightarrow \mathbb{R}^*$ 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{enumerate}

We give a negative answer to the problem posed by K.D. Li \cite{12}.