Selections and representations of multifunctions in paracompact spaces

Let (X,T) be a paracompact space, Y a complete metric space, ${\displaystyle F:X\to 2^Y}$ a lower semicontinuous multifunction with nonempty closed values. We prove that if ${\displaystyle T^{+}}$ is a (stronger than T) topology on X satisfying a compatibility property, then F admits a ${\displaystyle T^{+}}$-continuous selection. If Y is separable, then there exists a sequence ${\displaystyle \left(f_n\right)}$ of ${\displaystyle T^{+}}$-continuous selections such that ${\displaystyle F\left(x\right)=\overline{\{f_n\left(x\right);n\ge 1\}}}$ for all x ∈ X. Given a Banach space E, the above result is then used to construct directionally continuous selections on arbitrary subsets of ℝ × E.