Selections and sandwich-like properties via semi-continuous Banach-valued functions

Abstract

We introduce lower and upper semi-continuity of a map to the Banach space $c_0$($\lambda$) for an infinite cardinal $\lambda$. We prove that the following conditions (i), (ii) and (iii) on a $T_1$-space $X$ are equivalent: (i) For every two maps $g,$$h$ : $X\to c_0$($\lambda$) such that $g$ is upper semi-continuous, $h$ is lower semi-continuous and $g\le h$, there exists a continuous map $f$ : $X\to c_0\left(\lambda \right)$, with $g\le f\le h$. (ii) For every Banach space $Y$, with $w\left(Y\right)\le \lambda ,$ every lower semi-continuous set-valued mapping $\varnothing$ : $X\to {\mathcal{C}}_c\left(Y\right)$ admits a continuous selection, where ${\mathcal{C}}_c\left(Y\right)$ is the set of all non-empty compact convex sets in Y. (iii)$X$ is normal and every locally finite family $F$ of subsets of $X$, with $\left|F\right|\le \lambda$, has a locally finite open expansion provided it has a point-finite open expansion. We also characterize several paracompact-like properties by inserting continuous maps between semi-continuous Banach-valued functions.