A Strengthening of a Lemma on Continuous Families of Closed Convex Sets

Abstract

Let $X$ be a normed linear space and $E(\cdot )$ a family of set-valued mappings on a metric space $T$. lt will be shown the following strengthening of a result by S. Rolewicz. If $E(\cdot )$ is lower semi-continuous in $t_o$ and $Oc$ int $E(t_o)$, then $Oc$ int $E(t)$ holds for sufficiently neighbouring $t$ of $t_o$ too.