We introduce lower and upper semi-continuity of a map to the Banach space () for an infinite cardinal . We prove that the following conditions (i), (ii) and (iii) on a -space are equivalent: (i) For every two maps : () such that is upper semi-continuous, is lower semi-continuous and , there exists a continuous map : , with . (ii) For every Banach space , with every lower semi-continuous set-valued mapping : admits a continuous selection, where is the set of all non-empty compact convex sets in Y. (iii) is normal and every locally finite family of subsets of , with , 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.
"Selections and sandwich-like properties via semi-continuous Banach-valued functions." J. Math. Soc. Japan 55 (2) 499 - 521, April, 2003. https://doi.org/10.2969/jmsj/1191419128