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

抄録

We introduce lower and upper semi-continuity of a map to the Banach space c₀(λ) for an infinite cardinal λ. We prove that the following conditions (i), (ii) and (iii) on a T₁-space X are equivalent: (i) For every two maps g, h:X→ c₀(λ) such that g is upper semi-continuous, h is lower semi-continuous and g≤ h, there exists a continuous map f:X→ c₀(λ), with g≤ f≤ h. (ii) For every Banach space Y, with w(Y)≤λ, every lower semi-continuous set-valued mapping φ:X→ \mathscr{C}_c(Y) admits a continuous selection, where \mathscr{C}_c(Y) is the set of all non-empty compact convex sets in Y. (iii) X is normal and every locally finite family \mathscr{F} of subsets of X, with |\mathscr{F}|≤λ, 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.