Continuous selection for set-valued mappings

We study properties of set-valued mappings $F$ admitting a continuous selection $f$ which is a continuous $ϵ$-selection (from the set of $ϵ$-closest points) for the images $F\left(x\right)$ $(x\in X)$. This is interpreted as an $ϵ$-selection for continuously varying sets in a space with continuously varying norms. We deduce new fixed-point theorems from the results obtained. We also study geometric-topological properties of sets all of whose $r$-neighbourhoods possess a continuous $ϵ$-selection for every $ϵ>0$. We obtain a characterization of such sets. (paper)